In two earlier posts this week, we've covered the most basic approaches to time travel, so why not confuse the hell out of everyone and try to explain the notion of alternate timelines?
Back to the Future part II dealt with an alternate timeline in the form of Biff's distorted future (though technically all of BTTF, save for the opening of the first movie in 1985, happens in timelines that get altered by Marty's actions in the past.) The notion here is that there's one timeline and it is completely mutable. In changing things, you're really creating an altered timeline rather than an alternate timeline, despite the language Doc uses in Part II.
A truly alternate timeline would likely exist parallel to the "real" timeline. Thus, if you do something like going back to 1963 and saving JFK, you would have no impact on the history you just left. Instead, your changes would cause a new timeline to branch off from the old one, effectively creating an entirely new universe without wiping out the old one. I mention this because it can be a great way around the "paradox problem" one might encounter when writing time travel movies.
According to writers Roberto Orci and Alex Kurtzman, this is the approach they took to time travel in last summer's Star Trek. Star Trek is also novel because it's a time-travel movie set entirely in an alternate timeline, with the characters themselves not becoming aware of that fact until at least midway through the script. In other words, it's a time travel movie told from the point of view of the people who have been impacted by changes the others faced. Supposed that the assertive George McFly we meet at the end of the first Back to the Future learned that he was actually supposed to be a wimpy nerd married to an overweight alcoholic and you'll get a sense of what I'm trying to explain.
Star Trek's approach is smart because it found a way to honor over 40 years of existing canon without making it a flat-out reboot that erases everything to start fresh. The film starts when a Romulan ship from the future ends up in the early 23rd century. This triggers a battle with the Starfleet vessel Kelvin, which in turn causes Jim Kirk's mother to go into labor early. Not only that, in the course of the battle, Jim Kirk's father George Kirk ends up sacrificing himself and the ship to save the rest of the crew. This is significant because in the original timeline where this encounter didn't take place, Jim Kirk was born weeks later in Iowa, and his father George lived to see him become captain of the Enterprise 30 years later.
Thus, Trek history has been altered at the moment of Kirk's birth and everything that happens after this is an alternate timeline. This is confirmed later, when a now-adult Kirk is stranded on an ice planet and encounters a version of Spock from 130 years in the future - a Spock played by Leonard Nimoy, the actor who carried the role through the earlier incarnations of the series. This was smart on the part of the writers because not only does it allow them to say to fans, "That history you loved still happened. We're honoring it, not ignoring it," but it also makes clear that this new history that has emerged is going to be significantly different from that which is documented in the original series and movies.
To drive home the point, the planet Vulcan - a significant element in the original series timeline - is destroyed in the course of the film. Even though the film ends with all the classic characters in their familiar positions, it's clear that none of their fates will be the same as in the first series. Kirk could be killed in the next movie, if the creators saw fit.
The writers have said that the way they see it, the "original timeline" that the Nimoy Spock left still exists and marches on parallel to the new one. I'd argue that nothing shown in the film itself specifically supports that notion, but there's nothing that either disproves that either I suppose. Since it's unlikely we'll ever revisit the original, prime timeline again, it's probably a moot point.
I mention all this because it shows there's a way to use time travel without relying on the paradoxes, as Back to the Future does. Here, the time travel is a way of achieving and "in-the-box" reboot and giving all the characters a clean slate. So keep in mind that if you want to use time travel, you don't need to rely on the old "we have to put the future back exactly the way it was meant to be" chestnut. In Star Trek, the mission isn't really to restore the original timeline - it's to stop the Romulan Nero before he does further damage to this one.