Walt was later attacked by a polar bear in the jungle. He hid inside a Banyan tree while the polar bear rammed at him. During this encounter, one can allegedly hear the polar bear say "Walter" as it swiped at the tree. ("Special")
The Latin name for "polar bear", Ursus maritimus, was mentioned on the blast door map, implying that the bears were used during experiments on the Island. ("Lockdown")
STATED GOAL, REPATRIATION ACCELERATED DE-TERRITORIALIZATION OF URSUS MARITIMUS THROUGH GENE THERAPY AND EXTREME CLIMATE CHANGE
While chasing Dave, Hurley ran into Charlie and Mr. Eko and asked, "Did either of you see a guy run through here -- in a bathrobe, with a coconut?", to which Charlie replied, "No, but I did see a polar bear on roller blades with a mango." ("Dave")
Charlie told Locke that he used to get high and watch nature programs on the "Beeb" (BBC), where he learned that polar bears are very clever and "are like the Einsteins of the bear community." ("Further Instructions")
When Phil caught Jerry and Rosie dancing and flirting in the DHARMA security room, Phil said that Rosie shouldn't be there, because it was their watch. Jerry joked that the worst thing that could happen would be the polar bears figuring a way out of their cages. ("LaFleur")
When Pierre Chang found out Hurley knew about the dead body in the back of the DHARMA van, he threatened to assign Hurley to a job "shoveling polar bear feces" if he told anyone. ("Some Like It Hoth")
There was a Polar Bear skull in the makeshift crib in Claire's hut. ("Lighthouse")
The writers chose a polar bear attack rather than a boar attack in "Pilot, Part 2" because J.J. Abrams wanted to hint early on at the Island's unique nature. He suggested a genetically-enhanced animal (an idea that eventually produced the hy-bird), but producer Sarah Caplan suggested that viewers could confuse that with the monster. According to Damon Lindelof, he and Abrams suggested "It should be a polar bear!" in unison."It should be a polar bear!"
Originally, director J.J. Abrams had been vehemently opposed to using a CGI polar bear on the show. The first version had a stuffed prop bear shot out of an air cannon towards Sawyer as he pretended to shoot at it. This version was shown on the pre-broadcast commercial before the pilot aired. Someone posted a freeze-frame capture of this online, which according to J.J., looked "laughable and ridiculous". He conceded to giving the CGI version, which he and the production team liked much better, a try. This was the version that ended up in the broadcast airing of "Pilot, Part 2". J.J. remarked that he was surprised people would analyze the show in such detail. The two versions can be seen below for comparison.
Polar bears are excellent swimmers that have been known to swim as far as 200 miles 
The polar bear may be a reference to ironic process theory. From Wikipedia, a classic example is Fyodor Dostoevsky's quote from Winter notes on summer impressions: “Try to pose for yourself this task: not to think of a polar bear, and you will see that the cursed thing will come to mind every minute.” 
The well-known "white bear" psychological experiments were performed by Wegner. These experiments, wherein the subject was asked to speak randomly into a tape recorder but was told "not to think of a white bear", resulted in compelling findings about attempts to suppress strange or unpleasant thoughts. Trying to avoid thoughts of the bear, most subjects mentioned a white bear very frequently in their taped meanderings. Called "ironic process theory," this finding is thought to prove that attempting to suppress a mental image only makes it occur more frequently.