[AGarvin]

Special Relativity does most folks heads when they first suss it out. Time dilation is derived through what are known as the Lorentz Transformations. Conceptually it's not that hard to understand once properly explained, just difficult to explain on a message board. A few simple diagrams can make it much easier. I'll give it crack though.

Basically, it centres around the fact that the speed of light is the same for all observers. The stationary observer will see the light travel further than the person in the ship will for the same event. If the distance is further, but the velocity is the same, then the time taken must be 'longer', or dilated.

Imagine the person in the ship, we'll call him the spaceman, is holding a mirror up to his face, and our event is a beam of light travelling from his face to the mirror, a distance of d. According to our spaceman, the light ONLY travels from his face to the mirror, distance d.

Now the 'stationary' observer sees it slightly differently. Since the spaceman is moving with uniform velocity, our spaceman has moved, according to this observer, between the time the light leaves the spacemans face and the time it reaches the mirror, a distance of d2. So according to the stationary observer, the light has actually travelled d + d2. Even though he's seeing the light travel further, the velocity of light is the same for both of them.

Using insanely oversimplistic math, (not to mention non relativistic), and plugging in a few nice round figures to illustrate the point, you have: t = d/v (t=time, d=distance, v=velocity)

According to spaceman, d=200, v=100, so time=2 (200/100).
According to observer, d=400 (d+d2), v still = 100, so time=4 (400/100).

In a classical, or non relativistic sense, the velocity of light according to the observer would be increased by the velocity of the spaceman, so even though it travels further, it would do it with a higher velocity, so time integrity would be maintained. Not so with light. All observers record it as having the same velocity. So according to our spaceman, it took 2 time units for the light to go from his face to mirror, but according to the observer, it took 4.

For the purists, the actual Lorentz formula is:

t0 = t1/sqrt(1-(v/c)^2)

where t0 is time in your reference frame lasting t1 seconds in a frame moving at v.

Hope I havent rambled on to much (and all of this makes sense).

Cheers,
Andrew.