Time dilation is observed by someone not travelling at your speed. At close to the speed of light, if I observed you, for example, boiling an egg at that speed, to you it would take 3 minutes. To me it might take a thousand years. For this reason, in my books I prefer to call it time contraction - because mathematically that is what is happening.
There must be symmetry in this, so, somewhat bizarrely, you will also see time dilation in those you are moving with respect to.
Time dilation allows time travel - but only into the future.
EDIT: In 1916 I think, Einstein published the General Theory - he had realised that gravity must have an effect upon time also. So clocks in a gravity well (like Earth) run a little slower than those that are not. For GPS to work properly, there is a small time adjustment, because a clock in space will run a little faster than one on Earth.
EDIT 2: If you spoke to someone over radio, with them moving at a very high speed, there wouldn't just be a time delay - there would be some stretching of the signal. In audio terms, their voice would sound slowed down - I'd never thought about this!
EDIT 3: I have to admit that, at superluminal speeds, FTL, there is no theory. Not even a hypothesis, because at the present time it is regarded as impossible. You end up trying to obtain square roots of negative physical quantities etc., and it just doesn't seem to work.