A pro baseball player can throw a baseball at around 90 miles per hour (mph). If he were to do the same throw whilst standing on top of a vehicle moving at 60 mph, then the ball would be traveling at 150 mph. Now, if the baseball player where to stand still and use a torch, the light would leave the torch at 299,792,458 metres per second (m/s). If he then was on a vehicle moving at 542 m/s, and shone his torch, the light would not leave the torch at 299,793,000 m/s. It would leave at 299,792,458 m/s.
299,792,458 m/s is the speed of light. Actually, it's the speed of light through a vacuum, light travels slightly slower through the air (and slower again through water, this explains the rainbow. The red end of the spectrum slows down more than the blue end of the spectrum when light enters water, and so this is why we can see rainbows).
299,792,458 m/s is also the fastest anything can move in the universe. To achieve this speed, you must have no mass. The faster you want to throw a ball, the more effort (and therefore energy) you must put into throwing it. To move something with mass at the speed of light would require infinite energy. So far the photon (a particle of light) is the only known massless particle, though others are theorised such as the graviton.
This all might sound theoretical at the moment, but it is used practically as well. Without our knowledge of the above, Sat Navs would not work. If you were to go and drive with a Sat Nav for an hour, the satellites that work out where you are would actually experience less than an hour. This is pretty counter intuitive.
The best way that helped me understand it is like this: Imagine you're in a car going at 60 mph. If a car drove past you at 70 mph, from your point of view in the car it would like it was moving past you at 10 mph. Likewise, if you were level with another car travelling at 60 mph, from your point of view, they would look still (and this effect is used in action movies for our heroes or villains to jump from one vehicle to another whilst they move at high speed). If you then accelerated to 70 mph, it would like the car that was next to you was now moving behind you at 10 mph.
Now if we think back to the second paragraph, light always travels at the speed of light. In all the cases above, whichever of the cars you are in, you will always see light travelling at the same speed. Speed is distance traveled over a period of time. If light always travels the same distance over a period of time, no matter how much distance you are travelling in that time, something has got to give, and this is relativity, as described by Einstein. It's called relativity because time is relative. The faster you move, the slower time passes for you relative to an external observer who is moving slower than you (or not moving at all).
This effect is called Time Dilation, and was used in the plot for the original Planet of the Apes films - Charlton Heston et al travelled on a ship going at close to the speed of light, and were on it for 18 months, however, as the ship crashes onto an unknown planet, it's noted that the year is 3978 - a full 2006 years after the crew left Earth. The satellited servicing our Sat Navs also experience time dilation, and this must be taken into account when they calculate where exactly it is we are.
Now, it's true that our understanding of relativity is incomplete, as it doesn't (yet) tie in with our understanding of gravity. But this does not mean that it is wrong. Newton's Laws of Motion do not take into account relativity (which was why they caused a headache when trying to predict the orbits of the planets), but they were not wrong when described objects moving on Earth. Working out the flight path of a plane, or the airspeed velocity of an unladen swallow, Newtonian mechanics are just fine. We know that Einstein's Special and General Theories of Relativity are good approximations of the truth, because without them, things like Sat Navs wouldn't work, and it also explains a great deal more about what we observe in the universe.
If you've found this interesting, then I'd encourage you to check out the following books, as they go into great detail about the evidence for the above, and more besides:
Big Bang by Simon Singh < Especially this one, which I think is the most accessible
The Particle at the End of the Universe by Sean Carroll
Why does E = mc2? by Brian Cox and Jeff Forshaw
*The last post in the conversation said "Particles are still physical, so in theory, like waving your hand through the air, you would push those particles away from the point of origin faster than their original speed"