Wolverine is just like any other classic Marvel character - His name explains what his powers are. Just as Spider-Man has spider-powers, Iceman has ice powers and Iron Man has iron powers, so Wolverine has special wolverine-based powers. But considering that no-one (including his Marvel creators) is really that familiar with a wolverine's abilities, I thought it best to list Wolverine's powers here for you:
Enhanced Healing Ability
This is Wolverine's main power. It varies in strength depending on how powerful the writer needs it to be at the time but has, at various points in his history, brought him back from being vaporized in an atomic explosion and resurrected him from a single drop of blood. It has its downsides, though; for one thing, it has a tendency to build mental blocks to hide painful memories, and for another, once accidentally started turning Wolverine into something more beast than human. In case you're wondering, it's also the reason that a man born more than a century ago can look so young; it magically heals his body from the effects of aging.
Because wolverines are natural predators, Wolverine has increased sight, smell and hearing abilities (No, I know it doesn't really make sense if you sit down and think about it too much). Smell is the one most demonstrated in the comics - particularly when either tracking or a "I don't know, 'Roro, somethin' don't smell right" line is needed - but all three have been shown to be true at one time or another. In a slightly nonsensical moment, neither taste nor touch seem to be similarly amplified, and in fact - thanks to the healing power - both have been shown to be minimized when the plot demands it.
Enhanced Strength and Stamina
You can, again, blame this one on the healing power. With a metal skeleton, it only stands to reason that Wolverine's body has increased his strength; otherwise he'd just collapse under his own weight.
Those Famous Claws
Finally, something actually related to the animal wolverines. Wolverine's claws weren't originally meant to be part of him; his original creators intended them to be part of his gloves, but X-Men creator Chris Claremont felt that that took away the character's uniqueness, and wrote a scene early in the character's history that showed him using the claws while bare-handed. For years, it was assumed - even by the creators - that the claws were created during the same process that gave Wolverine his metal skeleton, but 1993's Fatal Attractions storyline showed Magneto strip the metal from his skeleton and reveal that the claws were actually natural bone, and part of Wolverine's original mutation.
That Metal Skeleton
Wolverine's bones have been coated in Adamantium, Marvel's fictional unbreakable metal, meaning that - in addition to having a healing factor - his skeleton is unbreakable as well. Despite being entirely unnatural, the Adamantium coating has a tendency to "heal" thanks to Wolverine's healing powers, for reasons that can only be described as "writers not really thinking things through."