In the theory of relativity, time dilation is a difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from a gravitational mass or masses.
An accurate clock at rest with respect to one observer may be measured to tick at a different rate when compared to a second observer's own equally accurate clocks. This effect arises neither from technical aspects of the clocks nor from the fact that signals need time to propagate, but from the nature of spacetime itself.
In layman's terms, it means that time moves slower when you're moving really fast.
WARNING: MASSIVE SPOILERS Edit
In Top o Nerae! Gunbuster, those fighting in the war against the Space Monsters, most notably the main character, Noriko Takaya, need to travel very far distances, and to do that, they need to travel at near-lightspeed. A lot. So whenever Noriko, or her dad, for that matter, came home from space, more time had passed on Earth than in the near-lightspeed craft she was living on, the Exelion. Because of this, while all the people she knew on Earth aged, she stayed the same, having only experienced a few months of time as opposed to several years. In the end, this difference ends up being much more substantial with Noriko and Kazumi being forced to leave themselves alone out in deep space for the good of humanity, with 12,000 years having passed by the time they finally reached Earth.