Tetris Attack is best described as a hybrid of Tetris and Bejeweled. The objective of the game is to line up the colored blocks into rows of 3 or more (thus making them disappear), and to chain combos together.
The end result depends on the game mode the player selects (Versus, Stage Clear, et al.). In Versus mode, you dump "garbage blocks" on your opponent's side, pushing them closer to touching the top and losing the game. Garbage blocks can be reduced to colored blocks if a block elimination occurs that touches the garbage block. Sound complicated? It's not once you have the game in front of you.
Tetris Attack is awesome because it encourages high-speed pattern recognition, hand-eye coordination, and to a certain degree critical thinking (which is a subject in which most people don't receive enough reinforcement).
Tetris Attack should be remade because it's an excellent fast-paced puzzle game that modern gamers would receive well as long as the programmers didn't botch it, and because I have not seen any other game quite like it.
While we're in the business of remaking the great, why not remake the truly great? There's as much of a market for Terranigma and Tetris Attack than there is for another remake of a fossilized turn-based RPG that you can find on virtually any SNES emulation website on the internet.