The console versions (for PS2, Wii, GameCube and PC) feature 3 different modes: a dancing mode which simulates Dance Dance Revolution with the player pushing the button in response to arrows on the screen, a fishing mode in which the player collects pebbles and shrimp as well as air bubbles in order to breathe, and a belly-sledding mode where the player sleds and collects the number of fish needed, sleds to beat a given time or races another character down the hill. In the Nintendo DS version of the game, the rhythm sequences use gameplay mechanics similar to those in Elite Beat Agents, though greatly simplified.
The game also features each of these modes for two players. In the dancing and swimming games, the players compete with each other, while cooperating with each other in the multi-player belly-sledding mode. Interestingly, the game features songs not in the film, such as "Shake Your Booty" from KC and the Sunshine Band for when Mumble hatches, and "I Will Survive" from Gloria Gaynor sung by Gloria when Mumble and her break up. These are for the dance modes of the game.
Source: Wikipedia, "Happy Feet (video game)," available under the CC-BY-SA License.