I'll try to explain some of it as best as I can.
The domestic leagues are comprised of a number of divisions, teams that finish at the top move up to the next division, teams at the bottom move down. The numbers vary from league to league. While there are no playoffs, in most leagues there is a cup competition that includes teams from a number of the divisions. The cup runs throughout the season and is usually a one game knockout formula. There are some exceptions to that, but that'd overcomplicate things.
The Champions League is comprised of the top teams from the UEFA member countries, which is basically Europe plus a few other nations that aren't actually in Europe. Each nation gets to send a set number of teams based on formulas. There are automatic spots and qualification spots, which basically decides what stage of the tournament you start at. The tournament is a mixture of 4 teams groups and 2 game total goal knockout legs.
There's also the UEFA Cup (although that name changes occasionally), which is made up of the next tier of teams in their various leagues.
As for contracts, it's pretty similar to hockey. I'm not sure about guaranteed v. non-guaranteed etc. but the general set-up is the same. IIRC transfer fees go to the club only, the player doesn't get a cut.
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