When physics turned around and labeled time as a dimension, it caught many of us off guard. It makes sense; we had just never conceived of it in that way before. Despite this concept having been around for a while now, popular science and the general public still has a little trouble getting the idea right. We have questions like "How can time stop? It would then have to stop for a certain length of time, injecting time back into the equation." These kinds of questions present a fundamental misunderstanding of exactly what it means for time to be a dimension.
In order to help everyone wrap their heads around this, I am going to draw parallels to dimensions we are familiar with: space. Time is a dimension just like space. We do not say that space moves, neither does time move. Space is the realm in which mass-energy moves. The difference between two positions in space is called distance. In time we call this same difference in position duration. Distance does not move. Mass-energy can, however, move a certain distance. In the same way, duration does not move--mass-energy can move a certain duration.
Using this reasoning, we should easily see that time does not truly stop. Time was never moving in the first place. Objects move through time just like they move through space. If an object, say a photon, were to achieve the velocity c (commonly known as the speed of light in a vacuum), then that object would cease to move through time relative to the plane of reference that observes its velocity as c. Time does not stop, but objects can stop moving through time. Other apparent paradoxes can be explained in as similar manner.
For further study:
General and special relativity
The Velocity c