The speed of light is constant. No matter how fast you are moving, you will always measure the speed of light to be the same. ie. It's constant in all frames.
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Ok, so if someone was throwing a ball at you 20 feet per second and you run away 8 feet per second, the ball is then coming at you 12 feet per second. Therefore, if you ran 20 feet per second, the ball would appear motionless to you (briefly).
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This is called a galilean transformation. In Newtonian mechanics, ie, what you are taught in high school, this is indeed the case. To find the relative speeds of two moving obects, simply subract their speeds.
However, it turns out that the Universe doesn't actually work this way. The Galilean transformation and the physics that you were taught in high school are simply an appoximation, albeit a very good appoximation. Once objects reach speeds of about 90% the speed of light, this approximation breaks down. If you want to find the relative speeds of two objects moving at near the speed of light, it is necessary to use a Lorentz transformation.
http://mathworld.wolfram.com/LorentzTransformation.html
http://en.wikipedia.org/wiki/Lorentz_transformation
The relationship between speeds is no longer simple. This goes against every intuition that you have. But remember, in the case that the speeds are much much slower than light, the Lorentz transformation simplifies to a Galilean transformation. Everything that you have every done, ever come across in your entire life occurs at speeds much much slower than light. Your intution is thus built on Galilean transformations, and do not carry across to relativistic (near light) speeds.
edit:spelling