Update - January 2004
A new law has been passed in France, lifting the requirement for EU Nationals to hold a Carte de séjour. It would appear that this message is taking some time to filter through all the French legal system, and if you want to work people are still being asked to produce their Carte de séjour. I'm sure it will take some time to settle down, and we still have the option to apply for our de séjour if we wish to do so.
EU Nationals who live in France are being advised to carry their Passport and proof of address at all times (which is harder to fit in your pocket than the Carte de séjour!).
For more information, check out the British Embassy website or read a full and varied discussion on the Living France Forum.
Previous information, which I suppose is now redundant:
If you are an EU citizen then all you need to be able to enter France is a valid passport. However, if you intend to stay for more than 3 months then you have to apply for a Carte de séjour de ressortissant de l’Union Européenne (EU Resident Permit) at either the Town Hall or Préfecture (police station). In order to do this you will need the following¹ :
¹Depending on who you ask will depend on the what you need to apply for your Carte de séjour. The above list has been compiled from a number of sources, including the UK's French Embassy website.
² This should be in the form of a contract of employment (if employed), proof of retirement income (if retired), proof of registration with a university (if a student), your marriage certificate (if you have married a French National) or finally proof that you can support yourself (i.e. bank statement)
You must apply before the 3 month period is up, and when you do you will be given a receipt as proof of your application. If you are approved, then you will be given a resident permit for 5 years, when can then be renewed to a 10 year resident permit at the end of this 5 year period.
According to Finn Skovgaard, you may be requested to provide translations of these documents, but this is not required. There should also be no charge for the Carte de séjour.