What Maze1125 said is correct; if a ship moves at approximately the speed of light (99% or so), then it will appear to take 4 years to get to a destination 4 light years away, from an observer at either end point (these measurements will typically be done by bouncing light waves off the ship; i.e. receiving communications from them). However, the crew on the ship will observe a much shorter time, but they will also measure a shorter distance traveled; the two effects pair up so they will measure exactly the same velocity on their ship as an observer stationary to the endpoints of the journey.
The_AC said:
This might be a dumb question, but does this paradox prove that acceleration is what causes time dilation (and causes the time to remain dilated until acceleration/decelleration occurs), rather than velocity? I'm asking this because the reason the Earth's frame is "correct" is because it isn't accelerating.
Acceleration does cause time dilation, but in a very complex manner; as I haven't done any work with General Relativity I don't understand the relationship fully. Special Relativity relies primarily on zero-acceleration situations (called inertial reference frames), and does exhibit time dilation. So, simple time dilation is based on relative velocities, not acceleration. It is probably better not to think of relativistic effects as something that 'builds up' as you accelerate, since the relativistic effects during acceleration don't match up well with zero-acceleration situations.