You're pretty much correct. Each country has a domestic league (i.e. Premiership in England, Serie A in Italy, La Liga in Spain). The winners of each of the main leagues, as well as some of the runners up (it's complicated but basically the stronger leagues get more teams entered) and some of the winners of smaller leagues (who normally playoff against the runners up beforehand) go into the Champions League, which is like the elite tournament for Europe. It used to be winners only, and a straight elimination tournament, but for cash and TV reasons it's now expanded to crazy proportions.
There is also the UEFA Cup, which is like the Champions League for teams who aren't quite good enough (i.e. the intial losers in the CL get put in there, as well as winners of smaller leagues who don't get in)
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