The UEFA Cup is a Cup competition that matches several hundred clubs together.  The competition begins in late summer and runs all the way through May.  Like the Qualification rounds of the Champions League, teams are injected into a particular round of the tournament according to seed.

One unique thing about the UEFA Cup is that certain teams eliminated from the UEFA Champions League are retained in the UEFA Cup.  These are normally the third-placed clubs in the First and Second Rounds, who are then inserted into the Cup at about the midway point.  The irony, of course, is that a team that does too well can be knocked out in the quarterfinals of the Champions League, yet a lesser club can do mediocre in the League and go on to win the Cup.  Don't believe that's ever happened, but it is an odd possibility to say the least.