In Europe, every country has its own league...pretty self explanatory.
Within each country's Football Association, there are typically multiple competitions...for example, in England, there is the Premier League, the FA Cup, and the League Cup (the latter include teams from all the different tiers of English leagues). In Spain, there is regular La Liga play, the Copa del Rey, and the Spanish Super Cup (again, the latter two include teams from the lower tiers).
All of these individual European country's leagues fall under the umbrella of UEFA, basically, the European Federation. They sponsor two major club competitions (Champions League and Europa League), as well as the Euros. Champions League is made up of the top teams in each of the domestic leagues, with the number of qualifying slots determined by a league power ranking...top 3 leagues get 4 qualifying spots each, next 3 get 3 spots, etc. Europa League is for the top teams that don't make Champions League. There's a group of teams in the CL that lose in the qualifiers that get sent down to Europa League.
All of these competitions happen simultaneously, with most mid-week matchups for domestic cup, Champions League, or Europa League play.
It's complicated as hell, but once you get into it, it's fun.