France has an original system with an executive branch headed by two officials: the President and the Prime Minister.
President of the Republic
Under the constitution, the President was originally elected for a seven-year term; this has been reduced to five years. There is no term limit. The President names the Prime Minister, presides over the gouvernement (cabinet of ministers), commands the armed forces, and concludes treaties. The President may submit questions to national referendums and can dissolve the National Assembly.
All his powers are subject to countersigning ("contreseing") by the Minister, except in a few cases such as the dissolution of the National Assembly.
In certain emergencies the President may assume special, comprehensive powers. However, in normal times, the President may pass neither legislation nor regulations, though, of course, if the Parliament is from his political side, he may strongly suggest the adoption of certain legislation, or request his Prime Minister to take such or such regulation.
In the original 1958 constitution, the President was elected by an electoral college of elected officials. However, in 1962, Charles de Gaulle obtained, through a referendum, an amendment to the constitution whereby the president would be directly elected by citizens. Given France's runoff voting system, this means that the presidential candidate is required to obtain a nationwide majority of non-blank votes at either the first or second round of balloting, which presumably implies that the president is somewhat supported by at least half of the voting population; this gives him considerable legitimacy. Despite his somewhat restricted de jure powers, the president thus enjoys considerable aura and effective power.
As a consequence, the President is the preeminent figure in French politics. He appoints the Prime Minister; though he may not de jure dismiss him, if the Prime Minister is from the same political side, he can, in practice, have him resign on demand (and it is known that Prime Ministers are asked to sign a non-dated dismissal letter before being nominated). He appoints the ministers, ministers-delegate and secretaries. When the President's political party or supporters control parliament, the President is the dominant player in executive action, choosing whomever he wishes for the government, and having it follow his political agenda (parliamentary disagreements do occur, though, even within the same party).
However, when the President's political opponents control parliament, the President's dominance can be severely limited, as he must choose a Prime Minister and cabinet who reflect the majority in parliament, and who will implement the agenda of the parliamentary majority. When parties from opposite ends of the political spectrum control parliament and the presidency, the power-sharing arrangement is known as cohabitation. Cohabitation used to happen from time to time before 2002, because the mandate of the President was 7 years and the mandate of the Assemblée Nationale was 5 years. Now that the mandate of the President has been shortened to 5 years, and that the elections are separated by only a few months, this is less likely to happen.