British citizens moving to France will need a valid passport, but no longer need a work permit or a visa, in line with other EU nationals.
(NOTE - British citizens from the Channel Islands and Isle of Man are not seen as EU nationals).

The situation has changed with the very recent expansion of the European Union, so British citizens no longer need to apply for an EU resident permit ("Carte de sejour") within 3 months of their arrival - but many still find them useful as a form of identification.

U.S citizens planning to move to France for more than 3 months will need to obtain a long-stay visa ("visa de long sejour"). These are issued in the States, at the French Consulate that has jurisdiction in your area. Those who are granted a long-stay visa must apply for a Carte de sejour within a week of arriving in France.