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To understand relativity, you must first examine the word--relativity. Einstein did not just pick this term out of a hat. Relativity involves comparing how one set of events compare to the same set of events observed by someone else's reference frame (a reference frame is, in a sense, someone's point of view or observation). The term relativity comes in when you compare two people's reference frames. The two reference frames will be relative to each other (my space is relativity to your space, my time is relativity to your time, and vice versa). But contrary to popular belief in Einstein's time, light is not relative.
Relativity has special effects that confine matter from doing some things. For instance, nothing achieve, or surpass the speed of light (186,000 miles per second, 300,000 kilometers per second). The reason for that being this: When something is accelerates toward the speed of light, the amount of energy it takes to further accelerate increases exponentially. Therefore, it would take an infinite amount of energy to get something to go the speed of light. There are more factors that contribute to this (more on that later), but that is the main reason.
Another property that relativity defines is something called "Time Dilation." Time dilation is a very hard concept to explain in depth. So here are the basics: Lets say I'm on a train that is traveling half the speed of light, and you are standing next to the tracks (oh, by the way, we are both wearing clocks around our necks). Now as you see me going by at a speed of 93,000 miles per second (it's a magic train; very fast!), you look at my clock and observe something strange. You notice that my clock is going slower than yours. But as I look at you through the window of my train at you, I also notice something strange. I notice that your clock is going slower than mine. How could this be? You see my clock going too slow, and I see your clock going too slow. Can we both be right? In a way, yes. When a object, such as a train, increases in speed, time slow down in that reference frame. And the faster an object increases in speed the more time slows down in that reference frame. But it does not slow down forever. When it approaches the speed of light, it slow way down and continues to slow down until it reaches the speed of light. When it reaches the speed of light, time stops. But that is all hypothetical considering it is impossible for matter to achieve the speed of light. But for arguments sake, lets say that matter can achieve and surpass the speed of light. What then? Well if the pattern continues, when something surpasses the ultimate speed limit (light) time will reverses itself. This leads to interesting time traveling possibilities. But, as of now, we cannot achieve the speed of light, let alone surpass it. Interesting side note: Particle accelerators can accelerate parts of atoms to 99.9999 percent the speed of light. That leads me to the proof of Time Dilation. In particle accelerators, scientists can smash electrons into targets, which produce high-speed particles called Mu Mesons. These high-speed particles live for only about 2.22 microseconds as measured by their time. But these particles have been recorded to live for 100 microseconds or more as measured by the scientist's time. The only way this can be true is for time dilation to be correct. The last effect that I will go over here is "Length Contraction." This effect sounds like its name. As something accelerates, it becomes smaller and come compact, or, in other words, it contracted. If we go back to the train example, and replace the clocks with meter sticks, we find the same paradox as the last effect. When you look at me from the railway track, it appears as if the meter stick is shorter than a meter. But when I look out the window, your meter stick looks contracted. The same paradox, different aspect of relativity. I believe the paradox is called "the twin paradox." And again there is no real answer to the question "Who is correct?" We both are. We can again examine the effects of approaching the speed of light. The faster one goes, the more they are contracted until that darn ultimate speed limit thing. When they hit the speed of light (keep in mind that it is impossible) they become infinitely small, infinitely dense, and infinitely contracted.
That's that. That is a brief summary of what relativity is and what it stands for. It gets about 100 times more complex when you add the math and the real physics, but the math and physics are needed to fully understand relativity. I have links (on the links page) to web sites that contain a lot more in-depth information than this brief synopsis.