It has already been described well enough, but I'll chip in anyway.
Each regular season of their own country's league, the teams try to finish in a qualifying slot for champions league or europa league. The next season they will play in their own regular league and the Euro competion they qualified for.
It continues every year trying to achieve Euro Tournament qualification for the following season.
Qualifying slots for this season:
Big 4
Serie A Italy
Champions League: 1st , 2nd
Champions League qualifying: 3rd
Europa League: 4th , 5th
English Premier
Champions League: 1st , 2nd , 3rd
Champions League qualifying: 4th
Europa League: 5th
Spain La Liga
Champions League: 1st , 2nd , 3rd
Champions League qualifying: 4th
Europa League: 5th , 6th
German Bundesliga
Champions League: 1st , 2nd , 3rd
Champions League qualifying: 4th
Europa League: 5th , 6th
Then maybe
French Ligue 1
Champions League: 1st , 2nd
Champions League qualifying: 3rd
Europa League: 4th
Dutch Eredivisie
Champions League: 1st
Champions League qualifying: 2nd
Europa League: 3rd , 4th
Europa League play-offs: 5th , 6th , 7th , 8th
Portuguese Liga
Champions League: 1st , 2nd
Champions League qualifying: 3rd
Europa League: 4th , 5th
Scottish Premier
Champions League qualifying: 1st , 2nd
Europa League: 3rd , 4th
Some more
Belgian Jupiler League
Champions League qualifying: 1st , 2nd
Europa League: 3rd , 4th
Danish SAS-Ligaen
Champions League qualifying: 1st
Europa League: 2nd , 3rd
Greek Super League
Champions League: 1st
Champions League qualifying: 2nd
Europa League: 3rd , 4th
Tippeligaen Norway
Champions League qualifying: 1st
Europa League: 2nd , 3rd
Those are some of many Euro Countries leagues. As you can see there is a heavy bias, in qualifying slots for the money spinning 'coefficient' clubs. Champions of Norway don't even get an outright place, yet 3rd from one of the main leagues automatically qualifies.
For me 'Champions League' should only contain the champions, to give the lesser Euro countries a chance to participate. Europa League would be sufficient for the 2nd and 3rd spots. IMO It's crazy that 4th place in the English, German, Spanish leagues could win the 'Champions' league the following season.