From the Super Mario Wiki
Polar Bears are ferocious, white bears that live on the Mushroom World's North Pole and in the Real World. Koopa Klaus had a run-in with one after being defeated by Mario, Luigi, Princess Toadstool, and Toad in The Super Mario Bros. Super Show! episode, "Koopa Klaus." Polar Bears are also implied to inhabit the Land of Ice, as Mario warned Luigi that he might get eaten by one if he waited around instead of coming along with Princess Toadstool and himself.
In Super Smash Bros. Melee's Adventure Mode, Polar Bears patrol horizontal platforms on the Icicle Mountain stage. On rare occasions, they may appear from Party Balls, but can easily be knocked off the screen with a few hits or one strong hit. In Super Smash Bros. Brawl, the Polar Bear appears once again, in the form of a stage gimmick. On the Summit, the Polar Bear will occasionally stand on the top of the stage, causing it to sink into the water. This will allow a fish to jump out of the water, and attack the players who are running in the water.
In WarioWare: Twisted there is a microgame in which the player controls Popo, the boy of the two Ice Climbers. He is holding his hammer with his two hands, and he can smash it down to either the left side or the right side. On both sides, either Topis and/or Polar Bears will come attack Popo. Popo will need to hit them with his hammer. If he hits them, they waddle off the screen.
A cameo appearance from a Polar bear is also seen in Art Style: PiCTOBiTS as one of the pictures formed.