The league
The league is the season-long competition against the same nineteen or so opponents, decided on points. This is the one that defines a club's standing, and the one relegation belongs to.
The domestic cup
A knockout tournament running alongside the league, open to clubs from every division. Lose once and you are out. This is where a fourth-division club occasionally beats a giant, and those matches are treasured out of proportion to their importance.
In England it is the FA Cup, in Spain the Copa del Rey, in Germany the DFB-Pokal.
If you follow U.S. sportImagine if a minor league baseball team could be drawn against the Yankees in a single elimination game, at the minor league team's own ballpark.
The European competitions
Three tiers, in order: the Champions League, the Europa League and the Conference League. Clubs qualify by where they finished in their own domestic league the previous season, so no club is guaranteed a place and none can buy in.
This means a club can be playing four competitions in one season, and that beating a Spanish club on a Wednesday can matter more to its supporters than a league match on the Saturday.