No, the FA Cup is a separate competition.
If you are a Premier League club you will play in three domestic competitions. Premier League, FA Cup and League Cup.
The League Cup is usually sponsored so takes the sponsors name. At the moment it is called the Carling Cup. This cup is played by all the 92 league clubs in England. There are four leagues: Premier League, The Championship, League One and League Two.
The FA Cup is open to a lot of entrants and starts with many qualifying rounds for amateur clubs. League One and League Two teams join in the First Round Proper, but Championship and Premier League clubs don't join until the 3rd Round of the tournament.
Both the league cup and FA Cup are knockout tournaments (i.e. the team that wins goes through to the next round).
The Premier League has 20 teams who play each other twice over a season, home and away. After 38 matches, the team with the most points are the champions. You get 3 points for a win, 1 point for a draw and 0 points for a loss.
The three lowest teams are relegated to the Championship. The top 4 go into the Champions League, then 5th and sometimes 6th go into the Uefa Cup.
Best place to find anything out about any of these competitions is Wikipedia, at least for the basics.