FTL drive

(short form of faster than light drive) Any form of propulsion that allows a spaceship to travel faster than the speed of light. Since to our current knowledge this is not possible within our universe, science fiction authors have come up with various concepts to avoid 100 year-long, boring breaks while the spaceship travels to the next star. All of them somehow circumvent the restrictions set up by Einstein's theory of relativity.

Hyperspace drive:

The spaceship is pushed out of the known 4-dimensional space into something called hyperspace, subspace, or similar , where either the theory of relativity does not hold or the light is faster or the way is shorter. The remaining problem normally lies in navigating there and avoiding to reemerge within a sun/planet.

Jump drive:

If activated at certain points in the universe (jump points) the ship will be thrown through some mysterious tunnel which shortcuts the known space (through hyperspace ?) to a far away point. May take some time or happen instantly. No navigation is needed while jumping, but one might have to find (and travel to) the jump points first. Other names for jump points are wormholes or stargates (the latter are an artificial construct). Happen to exist e.g. in Niven's "The mote in god's eye", in Brin's Uplift series and of course in Deep Space Nine.

Warp drive:

The ship remains in conventional space but somehow modifies space around itself (e.g. compressing it) which allows it to appear to travel faster than light to an observer outside the warped region. An alternative realization would be a drive which makes extremely short but frequent jumps through hyperspace so the ship appears to move FTL in normal space while in reality is moving only in hyperspace. Warp drives are of course used in Star trek.

Don't forget to add a write-up once you got one of these things working.

Other things needed to conquer the universe: FTL communication and astrogation

Another method of FTL travel is to "Fold" space. This means a device can bend space-time so that travel between two points is instantaneous.

This can be visualized is by thinking of the universe as a sheet of paper. Now make two points a good distance apart on the same side of the paper. Draw a line between the two points, this is the normal distance traveled. Now fold the paper so the two points touch, this is the distance traveled when you fold space.