Special relativity basic concept When someone travels really fast, do things seem to slow down around him or
do things go faster around him?
 A: When you are moving fast relative to another object, events on that object will appear to proceed more slowly. For an observer on that object, you are moving fast relative to them, and so they will see events on you proceeding more slowly. Each observer sees the other to be slowed in time, and there is no way to decide who is "really" moving. This last part is the fundamental cornerstone of relativity, that only relative velocity has any physical meaning.
A: The traveler himself notices no difference in time, the phrase "moving clocks slow down" being misleading in that respect. If observer B's clock is moving in observer A's frame of reference, B will record less time than A for any two events. B's clock is at rest in his frame of reference but is moving in A's frame. B's clock is recording what is referred to as the proper time. But everything will seem normal to B.
A: Here many answers have pointed out correctly that from the traveller's perspective, the outside world would move with a velocity opposite in direction and equal in magnitude which an outside observer would measure of the spaceship.
I'd like to add that all frames of references really are equal. I was going to write "would appear to move" in the above paragraph rather than "would move". This is because of the deep-held belief in our minds that some frames of reference are "better" and "truer" in the sense, they are easier to work with. This is certainly true but we should all remember this is a purely artificial choice and though the calculations would be much harder, in the end, the answers would predict the same thing as long as we stick to the postulates of SR. Once in a while, knowingly working with seemingly wonky frames will help us appreciate invariance across different frames more.
