The Wolverine (aka Logan) is set after the events in X-Men: The Last Stand. Jean Grey is dead, although she still appears to him in his dreams. He's still broken up about having to kill her.
He's living in hiding up in the mountains of Alaska or Canada. A Japanese woman named Yukio finds him and tells him he needs to come with her to Japan. It seems he saved a Japanese soldier's life back in World War II, and now the dying old man wants to thank him. He also wants to try and use Wolverine's powers to prolong his life.
Wolverine ends up being a protector to the man's granddaughter, Mariko (Tao Okamoto). She is set to inherit her grandfather's company and a bunch of Yakuza thugs want to kidnap or kill her. There is also some complications involving her father and a few other characters. It got a little confusing, and it seemed like too many characters changed from good to bad or vice versa too many times.
This is a slightly better movie than X-Men Origins: Wolverine. This time, they don't try to turn it into just another X-Men movie. The focus stays on Logan and what's going on in his head. He's trying to get over Jean Grey, and he also may be falling in love with Mariko. I was disappointed with the third act. Just like every other superhero or comic book movie, there has to be a big battle with a giant robot or something like that. This time, since we're in Japan, he fights a giant robot samuri. Or samuri robot.
I didn't love the movie, but it was enjoyable. However, I'm really excited for next year's X-Men: Days of Future Past. Make sure you stay for the credits so you don't miss the little teaser for the next movie.