What is the grandfather paradox?

You've seen it in "Back To The Future". You've seen it in "Futurama". You've seen it in the 2009 reboot of "Star Trek". It's the Grandfather Paradox (or variations upon it), where somebody goes back in time and changes something so big that it causes an impossible scenario. But what is the Grandfather Paradox, how would it work, and what would happen if somebody actually pulled it off?

The Ultimate Grandfather Clause
Here's the most straightforward version of the paradox. Jim Bob III builds a time machine and goes back about 60 years, to when Jim Bob I is a childless 20-year-old. He then shoots his grandpa dead in the street (Jim Bob III has some problems), meaning Jim Bob II is never born. But if Jim Bob II is never born, then Jim Bob III is never born. And if Jim Bob III is never born, then he never goes back in time and Jim Bob I is able to live a long and happy life. But then Jim Bob I goes on to have a messed-up grandson intent on causing paradoxes, and we're right back at the beginning.

There's one solution to this paradox that theorists keep coming back to, although it does have some problems of its own. It's called the Novikov self-consistency principle, and it basically states that any events that occur while traveling to the past are consistent with and identical to the events that occurred the "first time around." In other words, there's only one past, and if in the future you end up traveling back to it, then you were already there to begin with.

Confused? Actually, you've probably seen this the self-consistency principle play out already. In "Harry Potter and the Prisoner of Azkaban", Harry is saved when a mysterious figure summons the patronus of a stag to drive off a group of attacking Dementors. One trip through time later, he finds out that it was he who summoned that patronus. And in "Game of Thrones" [spoiler alert for season 6], Bran discovers that his acts in the present were responsible for Hodor's unique speech impediment forming in the past (we won't go into "how" since we're trying not to break down at work today).

These pop culture examples don't fix any paradoxes, per se, but they still show how the paradox would be fixed. The patronus that saved Harry was there the first time Harry experienced it, and Hodor became Hodor because Bran had gone back in time all along. In other words, you can't go back in time and kill your grandfather, because your grandfather was never killed in the first place. If you try, then you'll fail, because your grandfather never died in the past.
Different Strokes For Different Space-Time Continuums
Here's the thing about the Grandfather Paradox - it's only a paradox in certain versions of the space-time continuum. And until we actually succeed in going back, we're not going to know exactly how our continuum works. But here are a few possibilities.

Fixed Timeline

This is how the timeline shaped up in that episode of "Futurama", and it's the version that Novikov's principle most clearly addresses. In this version, what's past is past. "The ink is dry." So when Fry thought that he had altered the timeline forever by leading his grandfather to his death, he was actually just playing out how the timeline had already went...which happened to mean that he turns out to be his own grandpa. Ick. Moving on...

Dynamic Timeline

This is probably the most popular conception of how timelines work, even though it doesn't really make a lot of sense when you think about it. It's the kind of time travel that Marty McFly had to deal with. For him, the changes he made to the past were reflected in the present, which is how he nearly erased himself from existence by being more charming than his youthful father. But if the present changes when you change the past, then won't you change too? And if you accidentally erase yourself from existence, then you won't really have a chance to set it right. This is where paradoxes thrive, and where sci-fi writers just have to shrug and admit it's more dramatic that way.

Multiverse Timeline

This is the kind of timeline that goes down in the "Star Trek" reboot, and it conveniently sidesteps any potential paradoxes by saying that every trip to the past creates a new reality wherein that trip to the past occurred. The past of the reality you left behind is still set in stone, like a fixed timeline, but the new reality you've created can play out in any number of different ways. So Eric Bana's villainous Nero didn't have to worry about causing any paradoxes, since the universe he was affecting didn't have anything to do with the universe he came from. Sort of a "what happens in Vegas, stays in Vegas," but with even more lasers.