The main engines used on Galaxy class spaceships, like the USS Enterprise, on Star Trek.

The warp engines use some sort of matter/antimatter reaction to propel the ship through the galaxy at amazing speeds.

There are certain "levels" of warp. 0 being the slowest, 10 being theoretically the fastest. Nobody knows exactly what happens when the warp 10 boundary is crossed. If I recall correctly, there was a theory about inter-dimensional travel though.

Update: See below. General Wesc is obviously more of a trekkie than I am.

c (the speed of light) is 670,000,000 mph.

Old warp scale

1. c*1
2. c*8
3. c*27
4. c*64
5. c*125
6. c*216
7. c*343
8. c*512
9. c*729
10. c*1000
11. c*1331
12. c*1728
13. c*2197
14. c*2744

Warp drive works by accelerating from slower than light speed to faster than light speed in less than Planck time, therefore, never actually being at exactly light speed

There are two warp scales: the old one and the new one:

Old Warp Scale
See tribbel's writeup.

New Warp Scale

1. c*1
2. c*10
3. c*39
4. c*102
5. c*214
6. c*392
7. c*656
8. c*1025
9. c*1516

TOS uses the old one, TNG, DS9 and VOY used the new.

In Star Trek: Voyager (Voyager isn't Star Trek) they reached warp 10 which basically meant they were going infinite speed: they were everywhere in the universe at once. There's great debate about the use of warp 13 in All Good Things...

Special types of warp drive include Coaxial Warp, Quantum Slipstream, and Transwarp. Soliton waves also work for ftl travel, but are a very different concept.
For slower than the speed of light travel they use Impulse Drive.