In addition to the FIFA Laws of the Game rules that Matt outlines in his excellent answer, continental and national associations may have slightly different or supplementary rules for captains, although national associations tend to be in line with the continental association they belong to. In any case, captains are expected to be exemplars of on field behaviour.
The most recent regulations for UEFA are the Regulations of the UEFA European Football Championship 2010-2012, and what applies to captains is:
14.04 Before each competition match, each team receives a match sheet on which the
numbers, full names, dates of birth and, if applicable, the shirt names of the 18
players in the squad (23 players in the final tournament) must be entered, together
with the full names of the officials seated on the substitutes’ bench and on the
additional technical seats. The match sheet must be properly completed in block
capitals and signed by the captain and the head of delegation.
16.04 If the taking of kicks from the penalty mark cannot be completed because of
weather conditions or for other reasons beyond control, the results will be decided
by the drawing of lots by the referee in the presence of the UEFA match delegate
and the two team captains.
23.02 Protests concerning the state of the field of play must be submitted to the referee in writing by the relevant officials before the match. If the state of the field of play
becomes questionable in the course of the match, the team captain must inform
the referee, in the presence of the captain of the opposing team, orally without
In some national associations, if a player is cautioned or sent off and a suspension applies, the period of suspension is doubled if the player was captain at the time of the offence. Although I can't find a reliable reference for, I think that the doubling of the suspension time applies even if a player was acting captain at the time, receiving the armband after the regular captain went off field due to injury or substitution.
At club level, captains have various roles that can range from traditional to official:
- They lead negotiations with club owners and managers regarding internal regulations, bonuses, etc, and in some associations such regulations are only valid if signed by all captains,
- They are generally expected to act as the public face of the players, it's not uncommon to see captains taking full responsibility after a loss,
- They are generally expected to act as mentors to younger players, especially players in youth and reserve teams.