If you are confusing about new game modes, I'll tell you how to play them.
Versus game is the mode that you could play a match from 2-4 players with a traditional battle. You can either choose Puyo or Tetris to play with that type. During Versus, you make chains or lines will sends some garbage to your opponent, with each game type has a different set of sending garbages. You can also play Team vs. Team battles, which you could play with 4 players, 2 in one team, by changing the characters to join into a single team. However, a team with all players are not allowed to play.
This is the game where you start a match with Tetris/Lucky Attack and Puyo/Fever. Players need to preset chains and line clears to save nuisances for a later attack. When time is out, the nuisances are sucking up into a black hole, exploding for a namesake revolution. Players who have the less nuisances will have opponents' nuisances crashing the field. Once the field is completely scattered, that player is eliminated on play.
In this mode, Puyos and Tetriminos simultaneously rain down from the field. Make lines and chains during a small frame to add up to their counter, sending extra garbages to your opponents. Tetriminos will crush on Puyos if they landed on, and after the Tetriminos landed the next drop will return crushed Puyos on the fields, and garbage is an exception. Once a piece lands into an X the player is lost.