Have you ever felt that time was standing still. Well it was, if you were traveling at the speed of light. Otherwise, time simply slows down if you're moving quickly away from a point of reference. By how much does time slow? It depends on how fast you're going. The following chart depicts the relationship.
So, if you're traveling "with" me through space (that is your velocity relative to mine is zero) your clock will advance at the same rate as (1 x) the rate at which my clock advances. If you're moving steadily away from me at, say, one-half (0.5 x) the speed of light, your clock will advance at about 0.87 times the rate at which my clock advances. That is, when I look at my clock and see that an hour has passed since I last checked it, only about 52 minutes will have elapsed on your clock.
Just to put that in perspective, suppose you're traveling at the rather brisk pace of 2,000 miles an hour (about 3 x the speed of sound). That's about 3 millionths of the speed of light, at which rate your clock will advance at about 0.999999999996 times the rate of my clock. Hard to tell the difference, isn't it?
That's special relativity for you.