Concordats as international treaties
Concordats take the form of treaties between the Church government and a national government. This allows them, in most cases, to claim precedence over national legislation, and sometimes even over a country's constitution.
Concordats take the form of bilateral treaties between two governments. On party is the government of the country concerned, the other, an entity called the "Holy See", which is both the government of the Roman Catholic Church and of the “State of the Vatican City”.  In 1977 the Holy See put these under international law by signing the Vienna Convention on the Law of Treaties. 
Principles of International Treaty Law
International treaty law is based on five principles: free consent, good faith, that the treaty is binding, that a fundamental change of circumstances jeopardises it, and that it is better to try to maintain the treaty than to end it.  Once the Vatican has obtained a concordat, much moral stress is put on the principle that it is binding,  but no mention is made of the principle that fundamental changes of circumstances can nullify a treaty. And certainly not of the fact that dictators have frequently signed concordats to get Church support, yet when they are replaced by democratic governments, the Church expects to retain its privileges. (See "The right gets a concordat, the left, a modus vivendi".)
Countries are usually divided into 'monist' and 'dualist' legal systems. In a monist system international law is automatically included into domestic law. However, in a dualist system like the UK the general principle has always been that international treaties must be explicitly incorporated into UK domestic law by Parliament before they can be applied to an individual case. 
Stages of a treaty
Like any other treaty, a concordat must first be signed, which signals the intention to proceed with it, and then ratified, which confirms this and finally be supplemented by enabling legislation which turns it into law. Concordats must pass through these three stages of treaty-making:
Signature of a treaty, in many cases, does not result in any formal legal obligations for the particular country. However, it does require the state to refrain from any acts which would defeat the "object and purpose of the treaty". Signature is often an expression of general support by the country for the treaty's aims and the general commitment of the government to begin the domestic legal requirements for ratification. Of course, a government may take a long time to ratify a treaty (due to domestic political considerations, for example, […]). Treaties are normally open for signature for a certain period of time (as set out in the treaty itself).
Ratification of a treaty occurs normally as a second step after signature. This involves an exchange notes promising to abide by all of its provisions. A government can only ratify a treaty after it has completed all legal requirements for ratification of international agreements under its domestic law. For example, domestic law may require the approval of the legislature (one or both houses, if it’s a bi-cameral legislature) and/or it may be necessary for the country to enact implementing legislation in order for the treaty to have domestic legal effect. Once this process is over, the government deposits its instrument of ratification as set forth in the treaty [...]. 
Enabling legislation gives the Government the authority to enforce the concordat and is the final step. A simple example of this is a law stating that the concordat is approved as a treaty, and is incorporated into national law. Generally this "enabling legislation" is rubber-stamped.
In a theocracy like the Vatican, or in a dictatorship, like many of its concordat partners, ratification and enabling legislation involve no parliamentary approval. Therefore Vatican concordats with dictators can go through unopposed. (For more details on how to rush a concordat through these stages, see Eighteen tricks to get a concordat through.)
The framework concordat
In cases where the consent may be long in coming the Vatican has recently adapted for its own ends a type of treaty which was developed for quite different purposes, the framework treaty. This is normally a form of multilateral treaty establishing a structure that is built upon by future treaties of greater intricacy, such as protocols (amending or supplementary treaties) or national legislation.  However, it has also been adopted by the Vatican for recent bilateral concordats, where the gradual approach that this allows helps to surmount political opposition. This strategy proved invaluable in Slovakia, where the brief Basic Treaty set out the principles in abstract terms and it was only when the follow-up "Conscience concordat" spelled out their detailed implementation, that most people realised how this could influence their lives. 
A framework concordat also gives the Vatican the chance to include a target date for a more specific concordat to come. Thus the first concordat with Israel, the Fundamental Agreement, set out in Article 10.2.c a tentative timetable. Although this only mentions “aims” and is not legally binding, it has proven politically useful for the Vatican, as this enables it to put pressure on Israel. In fact, the papal nuncio in Jerusalem used this in order to publically accuse Israel of breaking its “promises”. 
International human rights treaties
Treaties were meant to settle international issues peaceably. They serve to end wars, extradite criminals and preventing overfishing and pollution. Treaties were not designed to circumvent separation of church and state — yet that is exactly what concordats do. A concordat uses the machinery of an international treaty in order to oblige a country to give privileges to one particular religious organisation inside its boundaries.
The Vatican also takes advantage of the fact that some international treaties are more than just bilateral regulation of contentious issues between two states: they can be multilateral commitments to human rights, as well. The pope has tried to paint concordats as a type of human rights treaty, one concerned with “religious freedom”. 
The advantage of casting a concordat in the form of an international treaty is that this generally overrides national legislation. Because a treaty is an agreement between several countries, one of them cannot, without good reason, simply walk away from the commitment. Thus, following nineteenth-century precedents, Article 27 of the Vienna Convention on the Law of Treaties clearly states that one party cannot invoke its "internal law" to justify breaking a treaty. This obligation is felt even more strongly in terms of human rights treaties: no country can simply decide to violate people’s most basic rights.
Concordats are clearly a powerful legal instrument. They enable the Vatican to use for its own ends the legal precedence often accorded to international treaties, in general, and human rights agreements, in particular. Unfortunately, human rights lend themselves to manipulation because they are not absolute: they are a bundle of trade-offs where giving up a measure of freedom in one area helps to ensures it in another. The Vatican can distort the whole system by proclaiming one right absolute and then letting it infringe on other rights. For example, the European Union human rights lawyers warned that the Slovak “conscience concordat” threatened important human rights, such as access to health care. This draft concordat manages this by defining "freedom of conscience" as "freedom of Catholic conscience", by which is meant, not the norms of most Catholics, but the formal doctrines of the Church. It then proceeded to declare the imposition of Church doctrine, defined as (Catholic) "conscience", to be an unrestricted "human right".
The irony is that such concordats now masquerade as the very thing they undermine — human rights treaties. And so that no one will again try to point this out, the Vatican is now trying to get the United Nations to change its definition of human rights in order to fit Church doctrine and the concordats which help enforce it! 
1. Muriel Fraser. "The Vatican’s triple crown: church, government and country", Concordat Watch. http://www.concordatwatch.eu/topic-823.834
2. "Vienna Convention on the Law of Treaties", Vienna, 23 May 1969. The Holy See signed it on 30 September 1969 and ratified it on 25 February 1977. http://treaties.un.org/Pages/ViewDetailsIII.aspx?&src=TREATY&mtdsg_no=XXIII~1&chapter=23&Temp=mtdsg3&lang=en
3. "Principles of International Treaty Law", (a superb summary), http://www.walter.gehr.net/frame2bis.html
On the complicated question of how changing circumstances may invalidate treaties, see Aziz T Saliba LLM, "Rebus sic stantibus: A Comparative Survey", Murdoch University Electronic Journal of Law, Volume 8, Number 3 (September 2001). http://www.murdoch.edu.au/elaw/issues/v8n3/saliba83.html
4. Giovani Lajolo, Vatican Secretary for Relations with States, 15 January 2004. http://www.concordatwatch.eu/showtopic.php?org_id=872&kb_header_id=825
5. Alasdair Henderson, "War remains inside the court room – Part 2: the Torture Convention", UK Human Rights Blog, 14 September 2016. https://ukhumanrightsblog.com/2016/09/14/war-remains-inside-the-court-room-part-2-the-torture-convention/
6. Katherine Hall Martinez, (Deputy Director, International Program Center for Reproductive Law and Policy, NY), "Convention on the Elimination of All Forms of Discrimination against Women", (accessed in 2005 on the site of the United Nations Development Programme, but no longer available). http://www.sdnp.undp.org/ww/lists/cedaw/msg00072.html
7. Foreign and Commonwealth Office [UK], "Glossary of Treaty Terms". http://www.fco.gov.uk/servlet/Front?pagename=OpenMarket/Xcelerate/ShowPage&c=Page&cid=1007029396068
8. Jarmila Lajcakova, "End of Women’s Reproductive Health Freedoms in Slovakia", March 2005. http://www.concordatwatch.eu/kb-1228.834
9. “In November, Archbishop Pietro Sambi — formerly the apostolic nuncio in Jerusalem […] — caused a stir by saying that Israel has consistently failed to fulfill its diplomatic promises.” “Israeli envoy promises: pact with Vatican soon”, Catholic World News, 4 December 2007. http://www.cwnews.com/news/viewstory.cfm?recnum=55127
10. “Pope seeks to defend church-state concordat in Italy”, Ekklesia, 5 Oct 2007. http://www.ekklesia.co.uk/node/5857
11. Pontificia Academia Pro Vita, "The Christian Conscience In Support Of The Right To Life", XIII General Assembly, Final Declaration,18 April 2007. http://www.vatican.va/roman_curia/pontifical_academies/acdlife/documents/rc_pont-acd_life_doc_20070315_xiii-gen-assembly-final_en.html