Say there are three cars, one parked on the side of the road, one driving past it and another one overtaking that one.
To the person sat in the stationary car, the car being overtaken appears to be going at 20mph, say. The car doing the overtaking appears to be going at 30mph. To the person sat in the car being overtaken, the stationary car appears to be moving backwards at 20mph, and the overtaking car is going forwards at 10mph.
Now, in that example, if you replace the overtaking car with some light, travelling at 300,000,000 m/s, both the moving observer and the stationary observer will observe it travelling at the same speed, not with a 20mph difference between them, as you might expect. In fact, whatever speed you are travelling at, as long as you're not accelerating or decelerating, light will appear to be travelling at the same speed - around 300,000,000 metres per second. The speed of light is always the same in any inertial frame of reference.
This causes all sorts of problems with Newtonian physics (ie they don't apply when speeds close to the speed of light are reached), and lots of nastily complicated looking things called Lorentz transformations are used to work out things like the time passed between two events for one obeserver in relation to another. Also, as objects approach the speed of light, they shrink along the direction they are travelling - the Lorentz Contraction - however as from the point of view of the object itself it is stationary and everything else is moving at the speed of light, everything around it will appear to contract.
Another consequence of the theory of relativity is that the faster you travel, the slower time passes for you (well in fact for the observer time appears to pass at a normal rate - everything else would appear to be happening very fast) with the normal example being if a rocket leaves earth at a speed very close to that of light and comes back 10,000 years later, it is possible that for the people on board, only one year has passed. If you want to go to the future, go very close to the speed of light for a while and eventually you'll get there.
Alternatively you could do what the rest of us do and just wait!
Anyway, I think I managed to cover the basics here without any maths... I dunno why they don't teach this in schools as it's much more interesting than learning about most other stuff in Physics...