While Tetris Attack has a backstory of Kamek and Bowser trying to enslave you and your friends, the game is essentially a straight-forward puzzle game bearing almost no resemblance to the actual Tetris game. Originally this game had nothing to do with Bowser, Yoshi, or any of his friends, but was actually "rebranded" to include these famous characters when the game was ported over to the United States.
In Tetris Attack, you need to think fast to match up colored blocks to erase them from your side to keep your block wall low. Adding to the complexity of this simple concept is the fact that if you are able to create a group of 4 or more blocks that will be erased at one time, you begin to send chucks of blocks over to your opponent's side. This back-and-forth struggle ends when one person's screen fills all the way to the top. There are additional modes that the game offers including a Puzzle Mode where you are given a set amount of moves to clear all the blocks off of your screen.
Tetris Attack was released in 1995 in Japan and 1996 in the US for the SNES.