A material body can't be accelerated past the speed of light. For example, a motion which appears to be a constant acceleration in the frame of a spaceship will look like the ship getting ever closer to the speed of light without quite reaching it in other frames. Unfortunately, I can't think of a nice way to explain why this must be the case at the moment (the standard explanation is that "mass increases with speed" but this explanation is unsatisfactory in lots of ways, not least of which is that by looking at things in that way then you are forced to also consider that the mass of a body depends on which way you try to push it).

However, there is another problem with moving faster than light - doing so enables the construction of time machines. It's often said that you can choose at most two out of the set {special relativity, faster than light travel, causality}. I've explained enough about special relativity in the Very Brief History of Time for you to able to convince yourself of this (perhaps...). It was one of the questions in the Midterm for the VBHT, and it is in fact possible to demonstrate it by using diagrams rather than equations (which is always a good thing).

All of this applies in little tiny regions in the curved spacetimes of general relativity as well as it does in the flat spacetime of special relativity. However, what people are mostly concerned about is not "faster than light" travel as such but just "convenient interstellar travel", by which I mean the ability to travel to another stellar system and back in a very short time as measured by the clocks left at home. GR allows this under some circumstances by the use of wormholes, which are just convenient shortcuts between the two stars. At no point does a spaceship traversing a wormhole travel faster than light. You can't get from Earth to Tau Ceti IV faster than light can using a wormhole - if you try racing with a pulse of light that also goes through the wormhole then the light will always get there first. Being able to outrace a pulse of light that takes the long route by using the short route is no more mysterious than my ability to get from my desk to the door of my room faster than a flash of light that I reflect off Mars.

As well as the engineering difficulties associated with keeping wormholes open there are also problems with causality. In GR, it's possible to twist spacetime around to such an extent that you can travel into the past even if you always travel slower than light. Such a spacetime is said to contain "closed timelike curves", which is just physics talk for "time machines". It's possible to manipulate a wormhole in such a way to produce these CTCs. Furthermore, there are theorems on topology change in GR that basically say that the process of making a wormhole in the first place also produces CTCs so that we'd be forced to rely on pre-existing wormholes which may not even exist.

Fortunately, GR is only an approximation to laws of quantum gravity that we haven't yet figured out and these may help on both counts. Firstly, they may make a configuration of wormholes collapse just before it becomes a time machine. Secondly, they may allow the production of wormholes out of the Planck-scale structure of spacetime without the associated problems of CTC formation. But on the other hand they might not...