I assume this question is about the national elections, and will answer this only. The election cycle lasts five years for presidential and lower House (it used to be different, which caused problems. More on that is for another question), and six for the upper House. Elections are usually held on Sundays.
Presidential elections are held around May, in a two-round system. In 2017, the dates for the two rounds were set, by the Constitutional Council: April 23rd and May 7th. The first two candidates in the first round compete in a run-off in the second.
Note: for territories located east of mainland France, elections are the previous Saturday.
Legislative elections (lower House)
The lower House of Parliament is called the National Assembly.
In each constituency, the same two-round system is used to elect the 577 (in 2017) representatives.
In 2017, voting takes place June 11th and 18th.
Legislative elections (upper House)
The upper House of Parliament is the Senate.
The senatorial elections are indirect, through an electoral College composed by a mixing of officials. Each department is attributed a certain number of senate seats, and the voting system differs according to the number of seats.
Half the seats are up for grab each three years, so Senators are elected for six years.
Do they have compulsory voting?
No. Voter turnout for the 2012 presidential elections was at around 80%, both rounds.
What vote tallying method do they use?
French voters have to register at the town hall of their residence city during the year preceding the election (you only need to do this once, unless you move to another city. You are now automatically registered to your known address - usually your parents'- when you turn 18, but this is a quite recent change). This puts you on the official lists, and you are assigned a voting station close to your place.
Once you're successfully registered, you receive by mail information about all the candidates, and where you should go to cast your ballot.
Paper ballots are mainly used, but electronic vote is slowly coming, depending on your voting station.
There is no early voting, nor voting by post (except maybe for French expats). If you are not near your voting station on election day, you can give a procuration, which is basically allowing someone else to vote for you.