The UEFA cup (Union of European Football Associations)was for the runners up,(and maybe also 3rd/4th/5th), from each of the national leagues, and was also a simple knock-out competition.
The winners of national cup competitions, such as the FA Cup, would compete in the European Cup winners' Cup the following season.
These formats changed in the 1990s
The European Cup became the Champions league, and moved from a simple knock out format to a set of mini leagues prior to the Q/Final stage, with a knock-out format from then on. The main reason for this was to generate more games,and make more money for the teams involved.
In a mini-league of 4 teams, a no-hoper is guaranteed 6 games (home & away) plus t.v. revenue, instead of being out in the 1st round.
The format was recently changed to have two rounds of leagues (even more games = even more money) prior to the QF stage.
It now starts with 64 teams in 16 leagues of 4 teams, with the top 2 (total 32 teams) going into 8 leagues of 4 teams, and the top 2 from each league (16 teams) producing the 8 Quarter final matches.
The Champions league unlike the old European Cup, is not restricted to national league winners, now including 2nd placed teams, and sometimes 3rd & 4th.Bayer Leverkusen reached this years' final without ever actually having won the German league in their history.
The inclusion of runners up/3rd/4th has detracted from the prestige of the UEFA cup because it now has fewer of the big-name teams in it. To compensate, the European cup winners cup was abolished, and the winners of the FA cup now compete in the UEFA cup.
The UEFA cup now consists of 3rd-5th placed teams and cup winners.
If a team wins both their national league and cup, then they take their place in the Champions league and the cup final runners up get the UEFA place.
Hope that wasn't too confusing!
Personally, I'll stick with Irish native games. You yanks haven't lived until you see a hurling match :-)