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At the risk of possibly confusing you more, I'll try to give the geometric answer (relativity is all about geometry, just not the usual euclidean kind) since its really elegant.
Think of measuring the distance between two points on an x-y coordinate grid in high school geometry. You use the Pythagorean theorem: r^2=(x-x')^2+(y-y')^2. The x and y coordinates can be rotated, etc, but r will always be the same. It is a "real" quantity, unlike x-x', which is an arbitrary construction.
In relativity, there are 4 dimensions. Time is added to the usual three (and is also defined rather arbitrarily). A spatial r as above is no longer "real" in relativity. It depends on how the coordinates are set up. There is another quantity that replaces it. Its called the proper time T^2=(t-t')^2-(x-x')^2/c^2-... or proper distance S^2=-T^2.
This looks very similar to the Pythagorean theorem, but that minus sign is crucial. All the wierd stuff in relativity comes from that little change.
T is a "natural" measure of time since it is independent of observer. Because of the minus sign, though, its possible for completely different points to have zero time difference between them (even if (t-t') is not zero)! This is what's happening as objects move closer to the speed of light. Using the above equation gives
(T/t)^2=1-(v/c)^2
The time we measure on earth is t, but the "light's time" is actually T. So T/t=0 or T=0. The light "sees no time pass." Since massive objects can't actually get to the speed of light, this equation is more relevant for v<c.
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