So I'll safely skip the whole "you run to the right, and if you jump on turtles, you can grab their shells" stuff and get right down to business. The game controls like the original NES game, with one button for jumping and one for running and/or fireballs. But if you shake the Wii Remote in the air, you'll perform a spin attack. If you push down while in the air as Super Mario, you'll butt stomp. And if you run up to one of the other players and shake the Remote while holding down the run button, you'll pick that player up. Maybe you're just trying to help by throwing other players across treacherous gaps. Or maybe you're just trying to throw them into pits. The choice is yours. Yoshi also appears in the game, and he can grab other players with his tongue and spit them into bottomless pits, too. Or other players can toss fireballs at Yoshi--if he eats them with his tongue, he can spit them out at enemies.
Suits, obviously, are a big part of the Mario experience. The propeller suit lets you hove and fly for short periods of time, and you get it going by shaking the Remote. I also saw a penguin suit, which lets you slide on your belly and toss ice crystals that freeze enemies into solid blocks. It also appears that you'll be able to turn into a miniature Mario, which is probably used for fitting through teensy gaps or something.
When you die in a multiplayer game, you're returned to the screen in a bubble. Another player has to pop the bubble to release you back into the level. You'll be able to shake the Remote to make the bubble float in the direction of the nearest player. As long as one player is up and running, the level will continue. At the end of each level, scores and performance are tallied to denote which player did the best.
The rest of what I saw felt pretty familiar. You run to the right. There are mushrooms to collect and Piranha Plants to contend with. There is an ice world. You can wall jump. Or, to put it another way, it felt pretty awesome.