Time and Relativity
Though referenced in H.G. Wells' The Time Machine (1895), the actual science of time travel didn't come into being until well into the twentieth century, as a side-effect of Albert Einstein's theory of general relativity (developed in 1915). Relativity describes the physical fabric of the universe in terms of a 4-dimensional spacetime, which includes three spatial dimensions (up/down, left/right, and front/back) along with one time dimension. Under this theory, which has been proven by numerous experiments over the last century, gravity is a result of the bending of this spacetime in response to the presence of matter. In other words, given a certain configuration of matter, the actual spacetime fabric of the universe can be altered in significant ways.
One of the amazing consequences of relativity is that movement can result in a difference in the way time passes, a process known as time dilation. This is most dramatically manifested in the classic Twin Paradox. In this method of "time travel" you can move into the future faster than normal, but there's not really any way back. (There's a slight exception, but more on that later in the article.)
Early Time Travel
In 1937, Scottish physicist W. J. van Stockum first applied general relativity in a way that opened the door for time travel. By applying the equation of general relativity to a situation with an infinitely long, extreme dense rotating cylinder (kind of like an endless barbershop pole). The rotation of such a massive object actually creates a phenomena known as "frame dragging," which is that it actually drags spacetime along with it. Van Stockum found that in this situation, you could create a path in 4-dimensional spacetime which began and ended at the same point - something called a closed timelike curve - which is the physical result that allows time travel. You can set off in a space ship and travel a path which brings you back to the exact same moment you started out at.
Though an intriguing result, this was a fairly contrived situation, so there wasn't really much concern about it taking place. A new interpretation was about to come along, however, which was much more controversial.
In 1949, the mathematician Kurt Godel - a friend of Einstein's and a colleague at Princeton University's Institute for Advanced Study - decided to tackle a situation where the whole universe is rotating. In Godel's solutions, time travel was actually allowed by the equations ... if the universe were rotating. A rotating universe could itself function as a time machine.
Now, if the universe were rotating, there would be ways to detect it (light beams would bend, for example, if the whole universe were rotating), and so far the evidence is overwhelmingly strong that there is no sort of universal rotation. So again, time travel is ruled out by this particular set of results. But the fact is that things in the universe do rotate, and that again opens up the possibility.
Time Travel and Black Holes
In 1963, New Zealand mathematician Roy Kerr used the field equations to analyze a rotating black hole, called a Kerr black hole, and found that the results allowed a path through a wormhole in the black hole, missing the singularity at the center, and make it out the other end. This scenario also allows for closed timelike curves, as theoretical physicist Kip Thorne realized years later.
In the early 1980s, while Carl Sagan worked on his 1985 novel Contact, he approached Kip Thorne with a question about the physics of time travel, which inspired Thorne to examine the concept of using a black hole as a means of time travel. Together with the physicist Sung-Won Kim, Thorne realized that you could (in theory) have a black hole with a wormhole connecting it to another point in space could be held open by some from of negative energy.
But just because you have a wormhole doesn't mean that you have a time machine. Now, let's assume that you could move one end of the wormhole (the "movable end). You place the movable end on a spaceship, shooting it off into space at nearly the speed of light. Time dilation (see, I promised it would come back) kicks in, and the time experienced by the movable end is much less than the time experienced by the fixed end. Let's assume that you move the movable end 5,000 years into the future of the Earth, but the movable end only "ages" 5 years. So you leave in 2010 AD, say, and arrive in 7010 AD.
However, if you travel through the movable end, you will actually pop out of the fixed end in 2015 AD. What? How does this work?
Well, the fact is that the two ends of the wormhole are connected. No matter how far apart they are, in spacetime, they're still basically "near" each other. Since the movable end is only five years older than when it left, going through it will send you back to the related point on the fixed wormhole. And if someone from 2015 AD Earth steps through the fixed wormhole, they'd come out in 7010 AD from the movable wormhole. (If someone stepped through the wormhole in 2012 AD, they'd end up on the spaceship somewhere in the middle of the trip ... and so on.)
Though the most physically reasonable description of a time machine, there are still problems. No one knows if wormholes or negative energy exist, nor how to put them together in this way if they do exist. But it is (in theory) possible.