this is an updated version of this comment over at Booman Tribune. Those of you that also wrote in that thread about other countries, please do not hesitate to make diaries here of your own comments there, it will certainly be worthwhile.

The President of France is elected directly by the population every 5 years (it used to be 7 years, but this was shortened - by referendum - in 2002) and has a lot of power - he chooses the prime minister, he can call for early parliamentary elections, and he has sole power to fill in a huge number of important positions in the administration, the judiciary and other. He is the Head of State and also, in normal times the head of the executive.

By normal times, I mean when he has a clear majority in Parliament: he chooses a prime minister from his side, who is formally the head of the government , but is often used as a lighting rod by the President, and he has a majority to vote his laws.

But the parliament can have a different majority, as has happened with increasing frequency in recent years (in 1986-88 and again in 1993-95 under Mitterrand and in 1997-2002 under Chirac). It's what's been called "cohabitation", as the two sides have to share power, but it's not really gridlock. In that case, the Prime Minister is chosen from the other side, and governs with the support of its majority in Parliament. The policies are clearly set by the Prime Minister, but the President keeps specific powers with regards to diplomacy, military affairs, and lots of domestic nominations - which thus require compromise between the two sides.

You basically have one major party on each side, the UMP (now led by Nicolas Sarkozy) is right of center, while the Socialist Party (currently led by François Hollande) is left-of-center. Both would probably fit inside the Democratic Party in the US in terms of policies, as there is a strong Statist streak in all parties in France. The UMP is mostly socially conservative, and more favorable to corporations and the wealthier classes, but does not really contest the French social advantages and is favorable to a strong State. The socialists are socially liberal and more favorable to workers and unions, but they are torn between an economically centrist wing (favorable to privatisation, free trade, and pro-European) and a harder left wing (which still hates the market economy, even if they don't call for a revolution anymore, and are often protectionist).

Alongside the two major parties, you have smaller parties on each side - the pro-European UDF (led by françois Bayrou) and the nationalist UPF (led by de Villiers) on the right, the Greens and the Communist Party on the left. Then you have the extreme right (Le Pen's National Front) and the extreme left (a bewildering array of trotskysts, the biggest groups being Lutte Ouvrière and the Ligue Communiste Revolutionnaire).

Some elections are base on proportional voting, and some are based on two-round majority vote (the top two candidates in the first round face off in the second round).
The two-round voting system used in parlimanentary elections (held every 5 years) leads to fairly stable coalitions, as the losing candidates of the left in the first round call to vote for the leading one, which most of the time comes form the socialist party (and the same on the right) - so the second round is usually a traditional left-right duel. In some cases, the parties on one side agree not to field candidates against one another in the first round to help the smaller parties go to the second round and get some seats.

Proportional voting, which is used for local elections, allows the smaller parties to get political appointees at the local level, and to measure their relative representativity (this will be used to negotiate a share of "safe" seats in the national elections as described above).

Of course, this stable arrangement has been disrupted by Jean-Marie Le Pen's populist and racist National Front, which polls about 15% of the votes. It is formally on the right, but the mainstream right, to its credit, mostly refuses to ally with the NF candidates, and both the mainstream left and the mainstream right support the other side vs the NF if it ever gets to the second round - so the NF has almost never gotten seats in the Parliament. (There is actually a twist in parliamentary elections: the second round takes place between the top two candidates of the first round AND any candidate that gets more than 12.5% of registered voters - so if you have a strong participation, you can get "triangulaires, or even "quadrangulaires". The National Front has typically been able to force triangulaires in 80 or so electoral districts - out of 577, although they have almost never won one).

In 2002, Le Pen got to the second round of the Presidential election by coming a smidgen in front of the Socialist Party candidate (it was Chirac 20%, Le Pen 17%, Jospin 16%) who lost out votes to a number of smallish candidates on the left (one communist, one green, two trostskysts, and two others) - everybody thought that Jospin was a shoo in for the second round, and nobody bothered to vote for him in the first round...

Anyway, the system is fairly robust: allows for good representation of smaller parties at the local level but creates stable majorities to govern at the national level. There is an unresolved tension between the Presidential election and the parliamentary elections, as both provide political legitimacy and power.