Light speed is going faster than the speed of light, but this causes a time shift where the people doing the traveling age slower than those not traveling at or beyond the speed of light. This allows people who do a lot of traveling to remain young while their great grand children grow old and die.
[This is best illustrated in the anime Gunbuster.]
In hyperspace or other similar dimensions just beyond real space there exist slipstreams caused by the gravities of planets, stars or other objects. Ships can use special sails or other equipment to surf on these currents to travel much faster than otherwise possible. Navigating by these currents can be much more difficult because they do not take straight paths, but wind around systems and stars. Plus if a ship's sails are damaged, it can become trapped, and lost forever.
[Voyager/Honor Harrington]
This time shift has inspired many different ways to overcome it so that the trip is made even quicker, and without
the time problem:
In hyperspace or other similar dimensions just beyond real space there exist slipstreams caused by the gravities of planets, stars or other objects. Ships can use special sails or other equipment to surf on these currents to travel much faster than otherwise possible. Navigating by these currents can be much more difficult because they do not take straight paths, but wind around systems and stars. Plus if a ship's sails are damaged, it can become trapped, and lost forever.
[Voyager/Honor Harrington]
Hyperspace is a separate dimension (also described in Space Geography). Ships that travel within it are able to reach destinations quicker than through real space. Hyperspace can be entered by opening a hole into it, if a ship can not do this on its own it must make use of gates to create openings for them.
[Babylon 5]
Similar to Hyperspace, but relies on speed to create something similar to a separate dimension for quicker travel, but objects that exist in the regular world, like planets or stars are just as real, so courses must be plotted to avoid them. Most ships that use this form of FTL have gravity detectors that automatically disengage the engines when they detect the gravity field of such an obstruction.
[Star Wars]
This is a process of making several rapid and continuous mini-space warps. The end result is a ship that is going extremely fast, and to an outside observer, the ship will seem to have a strobe effect.
A space fold allows travel by literally creating a fold in space. Matter at the starting point of the fold is picked up and transferred to the second point of the fold.
[Robotech, Yamato/Star Blazers]
Warp bubbles are formed by creating a bubble of subspace or some other dimension around a ship which allows it to move much more quickly through regular space. The bubble allows the ship to warp several of the laws of physics, generating faster than light travel.
[ Star Trek]
A warp skip is slightly more complicated than most warps. The distance warped is determined by the amount of power and inertia the warping ship has when it begins the warp. The faster the ship is going when it warps, the further it will go, like someone making a running jump, verses someone who stands still and jumps.
[currently unproduced Phoenix; the starburst on Starscape]
Worm Holes are tunnels that link one point in space to another. There is usually no visual way to locate a worm hole unless it is in use, and are normally discovered by accident. Useful worm holes are stable, meaning a clear path that doesn't change. Unstable worm holes shift positions at one end or the other randomly; or will shred anything that enters them.
Some theories suggest that Black Holes can act as Worm Holes, and others suggest that worm holes can be artificially created.
[DS9; Odin]