The FA Cup is really, in idea, that all the football teams in England, even the local lower league teams, play against each other to win the billing as top football club in England for that year. It's basically a bunch of knockout round of football with the winner moving on to the next round until there's two clubs left. There's 4 round of qualify for the lower leagues clubs to have a chance to play in the FA Cup. After that 7 round of elimination until the final match up. Premier League teams don't actually start play until the 3rd Round of elimination. In reality, by the semi finals all the clubs left are usually from the Premier League.
The Premier League is title is fought between the top 20 clubs based on the previous season's result. To win, it is basically to have the most points by the end of the season. You don't want to be part of the bottom 3 as you get relegated to the league below, while the top 3 clubs from the league below get promoted up to the Premier League.
The Champions League is the top football clubs in Europe fighting to be billed as the top football club of Europe. I'm not sure how the clubs qualify in the other European nations, but in England the top 3 clubs in the Premier League based on the points in the previous season get to be part of the Champions League.
The Europa League is the less cared about version of the Champions League as it's clubs that almost qualified for the Champions League playing against each other. The teams that end up fourth to sixth place in the Premier League, based on the previous season's points, qualify for the Europa League.
Hope that answered most of your questions.
