We think of Time as a steady, constant flow of events . Its unrelenting passage gives structure to our Universe in such a manner that a Universe without it is unimaginable. Yet Time flows differently for objects moving relative to each other. This is an artifact of the fact that nothing can move faster than the speed of light. My attempt at understanding this intuitively is as follows.
Take two objects 'A' and 'B' moving towards a shared midpoint, 'C'.
A => C <= B
Assume that A and B are both one light year from C and both are moving at the speed of light. It will take them one year to both arrive at C. However, although one would assume that they are moving towards each other at twice the speed of light, in fact they aren't because the speed of light is the maximum speed possible -- they are only approaching each other at the speed of light. If this is confusing, don't worry -- it is completely counter-intuitive, given our normal "common sense" framework.
The best way I've found to gain a somewhat intuitive understanding of this is to consider the basic equation:
distance = velocity * time
Note that since the distance involved hasn't changed but the velocity is different than expected, something funky must have happened with the time. Specifically, when the velocity of A and B is measured with respect to each other, it is half of what we'd expect, so the time elapsed in this context (with respect to these two objects moving towards each other) must be twice as much as we expected -- i.e., it is elapsing twice as slowly as we expected.
One of the really interesting aspects of this concept is that it shows how tightly Space and Time are bound together. Time actually flows differently for objects in relative motion to each other in space. In other words, one can't view Time and Space independently from each other -- hence, the concept of Spacetime -- space provides the first three dimensions and Time provides a fourth dimension.
References:
1. Einstein/relativity (TBD)