The UEFA Champions League is just that -- a league of national champions. Each of the UEFA federation members send their national champions from a given year to the UEFA Champions League the next season. Some countries get to send more than one, perhaps their second and/or third are included.
The UEFA Cup is a Cup competition that works the same way as the national Cup competitions, except that this only includes first division clubs from across Europe.
Qualifying for these competitions are directly related to two factors -- (a) where the team finishes in the first division and (b) which national federation it comes from. Each competition takes a fixed number of clubs, with the slots apportioned to the national federations. For example, Germany normally gets three Champions League slots and two UEFA Cup slots. Since the Champions League is the greater of the two, it is the first, second, and third place clubs that go to the Champions League, while fourth and fifth place join the UEFA Cup. Naturally, this favors the larger soccer nations, which makes sense.
UEFA manages the overall schedule for the competitions, assigning most of its competition dates on Tuesdays, Wednesdays, and Thursdays, leaving the weekends to the national federations to play their own leagues. These competitions do not go every week.
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