Premier League - League
FA Cup - knockout competition with every team in the English Football Pyramid. English Pyramid is something like 15 levels which includes professional leagues, semi-pro and full amateurs. Begins later in the year.
EFL Cup - knockout competition with every team in the English League system (professional clubs, levels on the pyramid 2-4 + Premier League)
Every country has some sort of format like this. Spain has Copa Del Ray, Germany DFB Pokal, Italy Copa Italia, etc.
European Competitions (UEFA inter country comps based on country coefficient) include:
Champions League - Premier competition composing of the top 4 teams from the Premier League previous season, and possibly 5 if an English team won the previous Europa League. This is why Spain has 5 teams in the CL this season.
Europa League - Secondary European cup competition. Includes teams 5 & 6 from the previous Premier League season. Leicester are in this one. Winner of the Europa automatically makes Champions League the next season.
Europa Conference League - New third tier competition that started only this season. Previously 7th placed team in the Premier League would have to go through additional qualifying rounds to get into Europa, but now they have a two-legged playoff for every team that qualified to get in this one. Lots of teams from weaker European leagues like Austria, Luxembourg, Cyprus, Belarus, Bulgaria, Gibraltar, etc will be in this one along with 7th place finishers from top leagues (England, Spain, Germany, Italy, France, Portugal). Winner of this comp will get into Europa for next season.
It’s not that complicated