In the near future the world is in ruins. The Sentinels have not only kill the majority of the mutants--they are killing regular humans. So Kitty Pryde sends the conciousness of Wolverine back to the past--to his body in the 1970's, where he has to try to change these future events. A big part of this is preventing Raven/Mystique (Jennifer Lawrence) from killing Dr. Bolivar Trask (Peter Dinklage), the creator of the Sentinels.
I liked this better than X-Men 3, not as much as 1-2 and First Class. In fact, I enjoyed the cartoon WOLVERINE AND THE X-MEN better than this flick. It's not bad, just predictable and about thirty minutes too long. However, the original comic book story is far more effective than this--and will make you want to reread the Chris Claremont/John Byrne classic X-Men story.