What is church-state separation (aka political secularism)?
Some opponents of church-state separation redefine it as “state neutrality” to allow their group, among others, to get state funding. Others try to discredit it by conflating separation of church and state with “atheism”. But it's a political, rather than a religious doctrine and its purpose is to help level the playing field in order to give a better chance for human rights.
“Secularism” has many meaningsFor brevity, “separation of religion and the state” is often called “secularism”. In the course of its historical career this confusing word has acquired a number of different meanings, most of which we can ignore.
♦ Let’s skip the mediaeval origin of the word, where “secular” meant parish priests, who were “in the world” (in saeculo), as opposed to the monks, who withdrew to a cloister.
♦ Nor need we concern ourselves with the muddled nineteenth-century meaning of the word “secularism”. It was originally coined to express the personal philosophy of one George Jacob Holyoake. He used “secularism” for the whole bundle of things which he felt would lead to human happiness, such as well-being, science and doing good. His word has survived, but not his definition.
♦ Let's also ignore those who try to conflate “secularism” with “atheism”. When atheists do this, it seems to be the result of conceptual confusion, but when some clerics identify the two it sometimes looks like an attempt to discredit church-state separation by implying that it is godless and wicked.
♦ Another aspect of this is the special pleading of atheist, humanist and other sectarian organisations when they try to define “secularism” as “state neutrality”. When they do this, they are urging that many different religious (or irreligious) groups get state support. This turns the idea of secularism on its head. In France the 1905 law that separated church and state said (in article 2) that the state “does not ... subsidise any religion”. But those arguing for “state neutrality” pretend that the state should, instead, subsidise every religion.
♦ And finally, let us beware when the Vatican talks of “healthy secularism” , “open secularism”  or “positive secularism”.  These weasel words reduce the idea of church-state separation to a mere rejection of a favoured church. This re-definition can extend to labelling the non-Vatican variety of church-state separation as “aggressive secularism”,  “militant secularism”  or “radical secularism”.  But this is just name-calling to try to discredit anyone who opposes sectarian groups getting handouts from the state — by implying that anyone who asks them to pay for their own missionising, whether religious or atheist, is a wild-eyed political extremist.
Be suspicious of anyone who wants to qualify “secularism”. It needs no adjective. It's totally straightforward: Secularism = separation of church & state, period.
The English language didn't adopt the French term, laïcité, as most others did, perhaps because the concept very concept of church-state separation was so alien. (Even today England has a state church and lets 26 of its bishops sit in parliament's Upper House.) The world language is now is stuck with the ambiguous term “secularism”.
This term has spread to India, which far more advanced, at least in theory, in terms of separating religion from the state. Secularism is India's national ideal, because it is seen as the most democratic way to unify a diverse nation. Since India contains one-sixth of mankind, this is an important anchor for the word in its political sense.
However, languages other than English tend to use a word with a different root. In Ancient Greece, the laos/laikos just meant "everybody". This meaning is preserved in the word “laymen”, which means everybody, as opposed to specialists, whether medical, legal or religious. Most languages use the local form of this Greek word to characterise separation of church and state: laïcité in French, laicidad in Spanish,  Laizität in German, laickość in Polish, Лаицизм (laitsízm) in Russian and laiklik in Turkish.
In France the separation of church and state was carried out in the name of human rights, and it certainly enabled them to flourish. However, secularism is merely a precondition for this and doesn't by itself guarantee a free society. Separating the state from the church only helps to protect people from domination by one powerful set of special interests.
That church-state separation is not enough is shown by the Communist countries and Turkey, all of which have practiced authoritarian secularism ― and also in the 2013 secular constitution of Fiji, which could open the door to this.
Communist states certainly had separation of church and state, (though many priests turn out to have spied for the secret services of Eastern Europe.) The Russian Communists were reacting in part against the power alliance between the Orthodox Church and the Tsars. And, of course, they also wanted to control religion in order to neutralise a competing ideology. The Communists' desire to free women from patriarchal control and everyone from church control, appears to have been the better to impose their own dictatorship, rather than from any regard for human rights.
Secularism in Turkey also had a political purpose. Called laiklik, from the French laïcité, it began as a break with the tradition of the preceding Ottoman Empire which was theoretically ruled by religious law.  Just as the French Republic feared a counter-revolutionary Catholic backlash, the Turkish one feared an Islamic counter-revolution, and tried to prevent that through a state agency to supervise and regulate the religious realm.  This makes “Turkey’s peculiar form of secularism ... less about separating mosque from state than keeping religion squarely under an official thumb.”  Indeed, the secular Turkish state violated religious neutrality, ignoring the minority Alevis, while funding only the majority Sunni clerics and teaching only Sunni religion in compulsory classes in the schools. And since Prime Minister (and later President) Tayyip Erdogan's party came to power in 2002, Turkish secularism, imperfect as it was, has been eroded in favour of Sunni Islam. 
Turkish human rights are also being increasingly eroded. Erdogan has rejected equality for women , claimed that contraception is treasonous  and used the failed coup against him in 2016 as a pretext for eliminating all political opponents.  The next year he continued to consolidate his authoritarian rule by shutting down 149 media outlets, closing more than 2,000 schools and universities, firing more than 120,000 civil servants and jailing more than 45,000 suspected dissenters. 
However, in addition to church-state separation, we also need a vigilant democracy if we are to ensure human rights.
Democracy supports the individual against the pressure of the group and the individual conscience against the dogma of the group. To do this, democracy counts the votes of individuals. It doesn't let unelected chiefs speak for the whole group, whether these are politburo members or “faith leaders”.
The French, who had to fight long and hard to achieve human rights, understand how precious they are ― and how secularism makes them possible.
The [French] Republic has always recognised individuals, rather than groups: a French citizen owes allegiance to the nation, and has no officially sanctioned ethnic or religious identity. …This view of citizenship is fundamentally non-discriminatory and inclusive. 
Secularism is an anti-monopoly measure
Secularism by itself doesn't guarantee human rights. Separating the state from the the church only insulates it from religious pressure, but not from other influences such as political ideologies or commercial interests. We could view secularism as the application to religion of “competition law”, (aka “anti-monopoly law” or “antitrust law”). Competition law is there to safeguard people's rights in their role as consumers, while secularism is there to protect people's rights as employees, pupils and citizens when they use state-funded services of all kinds.
Secularism even mirrors the three classic features of competition law  which are: to prevent (religious) monopolies or other forms of social control, to prohibit privileged agreements (like concordats) and other practices that restrict free (religious) competition and to ban abusive behaviour which would lead to a dominant market position.
There's nothing “ideological” about secularism: it's just a way of preventing the exercise of power where it doesn't belong: interference of the state in the religious affairs or vice versa.
In 2015 a British ruling caused controversy by finding a London mayor guilty of using “undue spiritual influence”. His election had been supported by a letter to The Weekly Desh signed by 101 imams. (#549)  This says that "Islamophobia ... has been made an agenda for voting and voters", citing as "proof" the sociological platitude uttered by a Labour councillor that "Religions divide people". The letter also implies that the charges of corruption against the Muslim mayor were religiously motivated and that without him at the helm no one would be safe to practise his religion in peace. This letter was found to be an attempt “to persuade Muslim voters that it was their religious duty to vote for him”. (#165) This ruling was based on a British law which bans undue influence over voters, including the threat of spiritual injury, to induce or compel someone to vote in a particular way.
The Court drew two conclusions:
♦ While clergy are entitled to hold and express political views, they cross a line if they seek to use the power and influence of religious office to convince the faithful that it is their religious duty to vote for against a particular candidate (#161) [...]; and
♦ The target audience matters – a distinction being drawn between a sophisticated, highly educated and politically literate community and one which is traditional, respectful of authority and possibly not fully integrated with other communities living in the same area. (#159) 
The second point appears to be a waffle intended to excuse the intervention of Anglican Bishops in recent elections. The Court actually justified this largely on the basis that nobody paid them serious attention (!) Naturally, the Church of England didn't like this. A priest argued disingenuously that no one had explicitly threatened anyone with "hellfire and damnation".  But they didn't have to. The threat is there in the theology: Those "cursed by Allah" are sentenced to Hell and the clerics' scare tactics served to remind the faithful that this penalty applied to anyone voting the wrong way.
Of course, the real reason why spiritual injury is illegal is because, as the Court pointed out (#160), it contravenes Article 10 of the European Convention on Human Rights. This says that freedom of expression and information must be "in accordance with law" and mustn't restrict what is "necessary in a democratic society". This is the true basis of secularism.
Quotes on secularism
I believe in an America where the separation of church and state is absolute.... I believe in a president whose views on religion are his own private affair.  (More on his views here)
Contrast this with what Irish PM John A. Costello said in 1951, "I am an Irishman second, I am a Catholic first, and I accept without qualification in all respects the teaching of the hierarchy and the church to which I belong." The problem was, not that he accepted this personally, but that he was prepared to impose it on the whole country.
Mario Cuomo, then-governor of New York State, 1984:
I protect my right to be a Catholic by preserving your right to be a Jew, or a Protestant or a nonbeliever, or anything else you choose. 
Jacques Berlinerblau, Director of the Program for Jewish Civilization at Georgetown University:
A secularist is a person who advocates the strict separation of Church and State. 
Joseph Lee, Times Educational Supplement, 2010:
Secularism is not the opposite of religion, but the opposite of theocracy. Freedom from religion is also the guarantor of freedom of religion, in a world of mutually exclusive, competing beliefs. 
“Secularism”, BBC online:
Secularists oppose religion or the religious being afforded privileges, which ― put another way ― means others are disadvantaged. ... Some secularists are actually believers in a faith. While they believe, they don't think that belief is a reason for special treatment. 
Thomas Jefferson, Virginia Statute for Religious Freedom, 1786, guaranteeing that no one may be compelled to finance any religion:
... No man shall be compelled to frequent or support any religious worship, place, or ministry whatsoever.... Danbury Baptists of Connecticut, to President Thomas Jefferson, 7 October 1801:
[We believe] that religion is at all times and places a matter between God and individuals ― that no man ought to suffer in name, person, or effects on account of his religious opinions ― that the legitimate power of civil government extends no further than to punish the man who works ill to his neighbours. 
Margaret More: [Father, that man] is bad.
More: There is no law against that.
Will Roper: There is! God’s law.
More: Then God can arrest him. 
Richard Gilyead, letter to The Guardian:
Tony Blair and Cardinal Cormac Murphy O'Connor deliberately conflate secularism with atheism. Atheism is lack of belief in gods. Secularism is a belief in equality in politics, education and law, regardless of religious belief. So when they refer to “militant secularism” and “aggressive secularism”, respectively, then they are implying that such equality of treatment is a bad thing. 
Senator Barack Obama, 10 July 2006:
This separation [of church and state] is critical to our form of government because in the end, democracy demands that the religiously motivated translate their concerns into universal, rather than religion-specific, values. It requires that their proposals be subject to argument, and amenable to reason. I may be opposed to abortion for religious reasons, but if I seek to pass a law banning the practice, I cannot simply point to the teachings of my church or evoke God's will. I have to explain why abortion violates some principle that is accessible to people of all faiths, including those with no faith at all. 
* Tugrul Ergin, demonstrating for secularism, Canakkale, Turkey, 2006-05-05, quoted by Anthony Shadid, “A Journey to Defend Turkey's Secular Ideals”, Washington Post, 2007-05-06. http://www.washingtonpost.com/wp-dyn/content/article/2007/05/05/AR2007050501180.html
1. "Benedict makes a case for 'healthy secularism'", National Catholic Reporter, 2008-09-12. http://ncronline.org/news/benedict-makes-case-healthy-secularism
2. "Catholic-atheist meetings end with Pope Benedict appeal to youth", Reuters, 2011-03-26. http://blogs.reuters.com/faithworld/2011/03/26/catholic-atheist-meetings-end-with-pope-benedict-appeal-to-youth/
3. "Pope sketches 'positive secularism' for Middle East", National Catholic Reporter, 2010-10-11. http://ncronline.org/blogs/ncr-today/pope-sketches-positive-secularism-middle-east
4. Tony Blair, see note
5. Cardinal Cormac Murphy O'Connor, see note
6. Benedict XVI, "Address to the Bishops of the United States of America", Consistory Hall, 2012-01-19. http://www.vatican.va/holy_father/benedict_xvi/speeches/2012/january/documents/hf_ben-xvi_spe_20120119_bishops-usa_en.html
7. See dozens of articles in Spanish on "la laicidad".
8. Burak Sansal, Ataturk’s reforms, All about Turkey. http://www.allaboutturkey.com/reform.htm
9. Omer Taspinar, "Turkey and secularism", Daily Times (Pakistan), 2003-09-16. http://www.dailytimes.com.pk/default.asp?page=story_16-9-2003_pg3_4
10. Andrew Finkel, "The Drumbeat of Ramadan", New York Times, 2012-08-07. http://latitude.blogs.nytimes.com/2012/08/07/tensions-in-turkey-during-ramadan/
See also Cemal Karakas, Turkey: Islam and Laicism between the Interests of State, Politics and Society, PRIF Reports (Peace Research Institute Frankfurt) No. 78, 2007.
11. Halil M. Karaveli, “Not a ‘slip of the tongue’: only one religion is recognised as the basis of the Turkish state”, Turkey Analyst, vol. 5 no. 10, 2012-05-14. http://www.silkroadstudies.org/new/inside/turkey/2012/120514a.html
12. “Erdogan the Misogynist: Turkish Prime Minister Assaults Women's Rights”, Spiegel, 2012-06-19. http://www.spiegel.de/international/europe/turkish-prime-minister-erdogan-targets-women-s-rights-a-839568.html
“Recep Tayyip Erdoğan: ‘women not equal to men’”, AFP, 2014-11-24. http://www.theguardian.com/world/2014/nov/24/turkeys-president-recep-tayyip-erdogan-women-not-equal-men
Erdogan may have political reasons for trying to force ethnic Turkish women to increase their birthrate of two children, in the face of the Kurdish average of 3.4. See:
“Turkish Women on Trial for Protesting Plan to Restrict Abortion”, Bloomberg News, 2012-11-06.
13. Ceylan Yeginsudec, “Turkey’s President Accuses Advocates of Birth Control of Being Traitors”, New York Times, 2014-12-22. http://www.nytimes.com/2014/12/23/world/europe/erdogan-turkey-president-says-contraception-supporters-traitors.html
14. “Turkey’s Post-Coup Crackdown Targets Kurdish Politicians”, New York Times, 2016-11-05.
15. Garry Kasparov and Thor Halvorssen, “Why the rise of authoritarianism is a global catastrophe”, Washington Post, 2017-02-13. https://www.washingtonpost.com/news/democracy-post/wp/2017/02/13/why-the-rise-of-authoritarianism-is-a-global-catastrophe/
16. Henri Astier, “The deep roots of French secularism”, BBC News Online, 2004-09-01. http://news.bbc.co.uk/2/hi/europe/3325285.stm
17. “Competition law in the United Kingdom”, Wikipedia. http://en.wikipedia.org/wiki/UK_competition_law
18. Erlam et al v. Rahman et al, Richard Mawrey QC, April 2015. https://trialbyjeory.files.wordpress.com/2015/04/judgment.pdf
19. Martin Downs, Spiritual Injury voids Mayor’s Election, UK Human Rights Blog, 2015-05-03. http://ukhumanrightsblog.com/2015/05/03/spiritual-injury-voids-mayors-election/
20. [The Rev.] Giles Fraser, “The Lutfur Rahman verdict and the spectre of ‘undue spiritual influence’”, Guardian, 2015-04-29. http://www.theguardian.com/commentisfree/2015/apr/29/lutfur-rahman-tower-hamlets-mayor-verdict-undue-spiritual-influence
21. John F. Kennedy, Speech to the Greater Houston Ministerial Association, Houston, Texas, 1960-09-12. http://www.bartleby.com/73/669.html
Kathleen Kennedy Townsend, “Sarah Palin is wrong about John F. Kennedy, religion and politics”, Washington Post, 2010-12-03. http://www.washingtonpost.com/wp-dyn/content/article/2010/12/03/AR2010120303209.html
22. Mario Matthew Cuomo, “Religious Belief and Public Morality”, delivered 1984-09-13, The University of Notre Dame, South Bend, IN. http://www.americanrhetoric.com/speeches/mariocuomoreligiousbelief.htm
23. Jacques Berlinerblau, “The God Vote”, Georgetown / On Faith, 2007-09-06.
He later changed his opinion to argue that this was a “misconception" and that secularism now means that means “the government can fund or assist religion; it just can’t play favorites.
Jacques Berlinerblau, “My Take: The five biggest misconceptions about secularism”, CNN, 2012-10-06. http://religion.blogs.cnn.com/2012/10/06/my-take-the-five-biggest-misconceptions-about-secularism/
24. Joseph Lee, “FEfocus Editorial - Livin' on a prayer? In its proper place”, Times Educational Supplement, 2010-10-22. http://www.tes.co.uk/article.aspx?storycode=6061349
25. “Secularism”, BBC online. http://www.bbc.co.uk/religion/religions/atheism/types/secularism.shtml
26. The Virginia Statute for Religious Freedom, written by Thomas Jefferson in 1779 and enacted in 1786. http://en.wikipedia.org/wiki/Virginia_Statute_for_Religious_Freedom
27. The address of the Danbury Baptists Association in the state of Connecticut, assembled1801-10-07. To Thomas Jefferson, Esq., President of the United States of America. http://www.stephenjaygould.org/ctrl/dba_jefferson.html
28. Robert Bolt, A Man for All Seasons, 1954. http://www.cooper.edu/humanities/classes/coreclasses/hss2/library/man_for_all_seasons.html
29. Richard Gilyead, letter to The Guardian, 2008-04-10.
30. Senator Barack Obama, “Politicians need not abandon religion”, USA Today, 2006-07-10. http://obama.senate.gov/news/060710-politicians_nee/ [This defence of secularism typically sketches the counterarguments. “On one occasion, he made a speech defending affirmative action that effectively articulated the objections to it. Rightwingers believed Obama had shown them deep understanding and respect. It was a mode of discourse that Obama would employ again and again [...]” http://www.guardian.co.uk/world/2008/nov/06/president-obama-story-kenya-to-white-house-part-two