The Quick Mini Spoilerific Version
In the movie, the age limit for people is 30. (In the books -there are three of them - it's 21; the books are also vastly different from the movies. In fact, other than having a protagonist named Logan and the concept of people being killed to keep the population down, there's virtually nothing the same between them) At birth, people have a crystal implanted in their palm. Every few years, it changes color until finally it starts flashing. When it does that, you have somrthing like a day to report to Carrousel before the crystal turns black.
They undergo a rite where, floating, they try to touch this giant crystal at the top of Carrousel. Lasers kill everyone. The legend is that if you touch the crystal, you get to either keep on living, or get another 30 years, or something. Nobody has ever done it, though; it's a lie, anyway. Later on, Logan realizes that he's never ever known anyone to be renewed.
Logan 7 is a Sandman. Sandmen are police who hunt down people who try to avoid Carrousel. There are rumors of a place called Santuary; if you can get there, you can live as lnog as you want. Logan is given a special assignment by the computer that controls the domed city; it artificially shortens his lifeclock (he's only 26 or so) so he becomes a Runner, and he's sent to find and infiltrate Sanctuary.
He follows the trail but discovers that if there ever was a Santuary, nobody has ever reached it because of an insane cyborg called Box. Box has killed everyone who's ever come down from the city. Logan and Jessica manage to destroy him and become the first people to ever make it Outside.
They discover the ruins of the former civilization and find a human living outside. An old human. They have never even seen depictions of people past 30, so he's fascinating to them. ("Those lines in your face.. don't they hurt?")
Logan and Jessica return to the City. Logan is captured and interrogated by the computer to find out the location of Sanctuary. Logan's mind shows that there isn't and never has been a Sanctuary and since this clashes with what the computer knows, it destroys itself, also destroying the city. The people gradually walk out into the sunlight and crowd around the old man.