Time Dilation
The most striking thing about general relativity is time dilation. It's immensely counter-intuitive, to the point that no science fiction on TV has ever attempted to work the science into the plot, preferring to keep an impossible 'universal time' by using wormholes or some unexplained FTL technology.

The basic idea is that two clocks travelling at different speeds will 'tick' at different intervals and thus report different times. If clock A is moving faster relative to Clock B's frame of reference, it will appear to Clock B that A is slow. The effect is significant enough that GPS satellites in orbit must compensate for the effect when synchronizing with clocks on the ground.

So any particle moving at c, the speed of light, does not experience 'time' at all. If one could travel at c, getting anywhere in the universe would be effectively instantaneous from the perspective of the traveller. Naturally, in other reference-frames millions or billions of years will have gone by.

But one can't travel at the speed of light, as the energies required to move a heavy object to c move towards infinity. One could travel at a decent fraction of c using an anti-matter drive, given certain constraints surrounding acceleration and deceleration. A poster on Slashdot runs the figures, and it's fascinating:

I did the calculations for an earlier post:

If you accelerate at 9.8m/s^2 for half the journey and -9.8m/s^2 for the second half of the journey (so that it's just like earth's gravity) then you would arrive at the planet after:

1.94 arccosh(n/1.94 + 1) years

For n=10.5 light years, this gives 4.9 years.

For other values of distance:
4.3 ly nearest star 3.6 years
27 ly Vega 6.6 years
30,000 ly Center of our galaxy 20 years
2,000,000 ly Andromeda galaxy 28 years

(For distances bigger than about a thousand million light years, the formulas given here is inadequate because the universe is expanding. General Relativity would have to be used to work out those cases.)

If you had an 100% efficient engine (using anti-matter/matter), the fuel required would be:

d Stopping at: M
4.3 ly Nearest star 38 kg
27 ly Vega 886 kg
30,000 ly Center of our galaxy 955,000 tonnes
2,000,000 ly Andromeda galaxy 4.2 thousand million tonnes

I find it fascinating that within a human lifetime (for the people in the rocket) we could travel to another galaxy.

(I'm a theoretical particle physicist)

One problem is that at such speeds the 'friction' (note: not really friction, but a force analogical) from microwave background radiation would just disintegrate the ship.

Heck, bumping into a dust mote at .9998% of c would be catastrophic. But, man. 28 years for another galaxy? I'm in.

iopha
Bonus: Dozens of slashdot posters bewildered by time-dilation assert that duration is a hard constraint even for high values of c and thus that such figures are wrong somewhere.