I'm not a Catholic or a diplomat, and my reading of history is pretty good, but I had never heard of a Concordat until recently, when I came across Concordant Watch.
It can be as simple a treaty, or legal document that is written up between a sovereign state and the Vatican. It's one of those documents that both parties must agree to dissolve, but as it's usually about the Vatican keeping it;'s privileges then it's not likely they will do so.
The Vatican is a legal oddity, it claims to be a state, and the Pope to be the equal of a Prime Minister, or a country's President. the discussion usually is split by people who are Catholic or sympathetic who say it's a country, and those who are irreligious, and think that it's probably not. I haven't the slightest idea who is legally correct, I just find it seems to be what ever is convenient for the Vatican to be what it want to be at that particular time. In Italian law the Vatican is it's own microstate, this was confirmed in 1929 when the Lateran Treaty was signed between the Vatican and the Mussolini Government.
One thing to keep in mind is that the catholic Church isn't really big on Civil law, you know the stuff everyday people have to worry about. It's got it's own Canon Law, that it believes overrides Civil law. That's why raping a child will just get you moved to a different parish, but making a woman a priest will get you excommunicated.
So how does the Church really see it self?
The Catholic Church, for her part, has always maintained that she is a supranational community and does not receive her legal existence from any temporal State. The Catholic Church exists as such, and it is in the Holy See that her organic unity is guaranteed, represented and perpetuated. Her extension corresponds to the spread of Catholic believers throughout the world. She is therefore a universal community that knows no national frontiers . Put in other words, when a state and the Holy See conclude an agreement they enter into a relation as two "perfect societies": the first as a temporal society, the second as a spiritual one. The Church receives its laws from her 'Founder' and not, like states, from some temporal power. But that doesn't mean that acts of the Catholic Church can have no effect on temporal jurisdiction. An intersection is possible and at this point the sovereignty of the Catholic Church stops and the problem of concordats begins.
Why does this really matter?
The Catholic Church is a big employer, it has a million people working for it in Germany, and it is one of Australians biggest employers, one of the biggest landowners, and would be in the 10 richest corporations in Australia if is was actually incorporated, and could sort out it's accounts. In concrete terms it can mean that you can be discriminated against for employment, in a church run business, that is paid for by the state, such as happens in Australia, for not being the "correct" religion. Italy still pays the Vatican money for the loss of the Papal States. State funded, but church run hospitals will refuse abortions.
To to find out what has been happening in modern times we can use a Slovakia as an example. Oddly it has recently been turning more Catholic, a trend that has been a bit disturbing to the rest of the EU, where Poland and Ireland have been becoming more secular.
How have they been using in the past? Well it shows very simply that the German Government under Hitler was very keen on the Vatican, and signed the Reichskonkordat not surprisingly as he was a Catholic, Germany still continues to sign them to this day
Are they still in use in this day and age?
On the third of May 2011 Azerbaijan the and the Vatican signed a new one
East Timor is currently in the process of signing one.
To find out more about concordats this very very excellent page on Wikipedia sums the problems very well, and is far better than I could have written myself.
As usual with matters of the Vatican, caution must at all time be taken to recognize actual issues, and those that are products of conspiracy, or fantasy.
“Each concordat marks a state’s renunciation of its own powers and its assumption of the obligation to contribute to those of the Catholic Church. In exchange for these gifts, authoritarian governments buy from the church hierarchy a kind of legitimacy and support for their power, as being in harmony with the commands of God. In some cases they gain influence in the appointment of senior Churchmen. A democratic system does not need this legitimacy [...] To grant the Church any privileges violates the foundations of democracy, even if the facade is retained.”
— Anonymous Polish MP to President Kwaśniewski, cited by MP Ryszard Zając on 12 September 1996 in the Sejm.