There are 2 parts to the Theory of Relativity: Special and General. Special relativityonly deals with objects in uniform motion, while general relativity deals with accelerated objects, and is a much harder subject for that reason. According to special relativity, even for objects in uniform motion, strange and unintuitive things can happen. If there was a passenger train inside which, through the windows, you could see the passengers and their watches, and it whizzed by you, a stationary observer, the following things will happen 1) the train will have shrunk length-wise, and 2) the watches of the passengers will appear to be going more slowly, and 3) the watches of the passengers in the back of the train will be AHEAD of the watches of the passengers in the front of the train. When you get to general relativity, more wierd things can happen. Time and space can be warped, and for this reason we have gravity. That is the reason why Einstein's general theory of relativity is often referred to as his theory of gravitation. General relativity explains why the universe is expanding, and why there are bizarre objects like black holes, which are examples of extremely warped spacetime.
It almost doesn't need to be said that relativity says nothing can travel faster than light, and that E = mc², which is the reason why we have atomic bombs.