I went into The Wolverine knowing almost nothing about it other than it had Hugh Jackman playing his most famous role and we'd see the late Jean Grey in it. For all intents and purposes you should consider this X-Men 4. Sure, there's just Wolverine, but it picks up some years after X-Men 3 and continues the story.
Having killed Jean Grey, one of his great loves, he heads off into the wilds of Canada to live as a hermit and torment himself with her memory. He's drawn out when a bear that he knows gets killed by some hunters. While going after the hunters he meets a young woman who wants to take him to Tokyo where a man he saved from the Nagasaki nuke is dying. Things start going hinky almost immediately as a major corporation is preparing to change hands within the family, but not the way anyone expected. The story follows the standard Japanese lore about a lone samurai falling for a princess and fighting ninjas and whatnot to save her. Only this time there's mutant abilities involved and the big samurai who must be confronted at the end of the story is a robot.
While I enjoyed the movie and am glad that I saw it, I'm not feeling inclined to buy the DVD. What you really want to see the movie for is the short bit that comes during the credits. I'm not sure if it's leading into the new First Class movie or another sequel to the original X-Men movies, but it should get you excited.