The light clock is a tool that makes it easy to see how movement and time are tied together.
At the top and bottom of the clock are two mirrors that bounce a beam of light back and forth.  Time is measured by how long it takes light to make the trip up or down.


Your own clock (left), of course, always keeps perfect time.  But the clock of someone moving past you (right) runs a little slow because the light has a greater distance to travel between ticks.

Of course, the other person sees their clock working perfectly, and your clock running slow.  The quickest way to explain away this paradox is:
Since you're moving apart and you'll never see each other again: Who cares?



Technically...
To find the exact difference between how fast you clock tick and how fast the moving clock ticks is found by looking at how fast the light beam is moving up and down.
The Beam's total speed is always C.  The horizontal speed is V, the speed of the clock as a whole.  So its vertical speed, which dictates how fast the clock ticks, can be found with the Pythagorean theorem.

The length of the moving tick compared to the stationary tick is:

So for every tick that your clock makes, the moving clock makes 1/ .

In general I'll make moving time and position as T' and X'.  So here T' = T, since T' is slower.


 note:

This is a graph of = (1-(V/C)^2)^-1/2, with C=1.
Notice that  ≥ 1, and that  is almost 1 until V is very high.
For comparison, on this graph the speed of sound would be found on the x-axis at about V=0.000001.