If you're a fan of the puzzle genre, you can't go wrong with Polarium.
Polarium has a Tetris mode, but the REAL game is in the 100 puzzles. With these puzzles, you can take as long as you want, but you'll NEED that amount of time, because these puzzles are no pushover, like the Tetris mode is. Get ready to use Gamefaqs a LOT in this game; the puzzles will bend the brains of even the smartest gamers.
Graphics? Who needs 'em! This is a PUZZLE game! You forget: Puzzle games put a black-and-white handheld gaming device on the map! Puzzle games don't NEED to have good graphics... and, unlike most modern puzzle games, like Mr. Drill, this game doesn't have a story, but it doesn't NEED a story, either!
Now, these 100 puzzles already provided are all fine and good, but you can have nearly infinite replay value using the level editor. This game isn't online, per se, but it has a code for every possible custom map you can possibly design. When you design the map, you can swap the code. This adds a whole new area of fun to this game: Just type in a random code and see what you can come up with.
For four years, this was the best puzzle game on the DS, and it might still be the best, but I think Scribblenauts can give Polarium a run for its money.