Dean is way off base. Time on board the rocket actually slows down relative to an outside observer who is fixed relative to the ship. If we could see it from Earth, we'd see your $5,000 Rolex lose time and the faster you went in the rocket, the more it would lose time relative to our clocks on Earth.
This not only affects your Rolex, it also affects your heartbeat rate, which also slows down. In fact, your entire basal metabolism will slow down; so you will not age as fast as that person on Earth.
On the other hand, if that outside observer were to climb into another rocket and come up along side yours, so your relative velocity re his ship is zero, then you would see his Rolex and your Timex would be in sync. That results, because you are at rest relative to him and he is at rest relative to you in this case. So between the two of you, time marches on as normal rest time.
All this time slowing stuff is called time dilation. It is a major result of the general and special theories of relativity. Most of the time, we don't experience time dilation. It's there, but it's so small we just don't see it.
But significant time dilation happens when relative velocities are extreme, like with the rocket. This is described in the special theory.
It also happens when that Rolex is subjected to extreme gravitational force, like in a black hole. This is the effect of the general theory, which describes spacetime and how it bends under extreme acceleration, like from gravity.