There are 5 ways to play a game on this website:

• Monthly Tournaments (Covered in the FAQ, no problems there)
• Monthly Cup (Not covered)
• Ladder (Not covered)
• Rating Games (I assume the same as in Waiting Room?)
• User Tournaments (Simple enough)

Can someone explain how do Monthly Cups and Ladders work, and the difference between them and Monthly Tournaments?

The FAQ covers only the Championship tournaments (not Monthly tournaments, I guess that was a typo on your side).
They are not monthly but a new championship tournament starts for a game when the current tournament finishes - and that means all of the leagues (sub-tournaments) finish.

About the other types of tournaments, see the excellent FAQ by Ed Collins

• Monthly Cups
• Infinity/Ladder
• Rated games (waiting room)
• User tournaments

Yep it was a typo, thanks for the link