Think about some fluorescent bulbs. When you first turn them on, they're still kind of dark, and only after they get warmed up do they reach their maximum brightness. LCD's are more or less the same -- when changing to a color, it sets the voltage to what's appropriate for that color, and waits for the liquid crystals to reach equilibrium. Unfortunately this takes a while, and the time it takes to reach equilibrium is the response time measurement (or at least, is supposed to be; manufacturers use their own definitions that's most advantageous to them). The LCD part itself doesn't actually have any color; the red/green/blue comes from a color filter layer. So this whole white/gray/black business has to do with the intensity of the light that's let through, not the color. White means maximum brightness (for any color), black means minimum brightness, and gray means anywhere in between.
For LCDs, it's actually the gray-to-gray transitions that take more time. The reason is that the speed that it turns is higher for bigger transitions. Thus, going from 128 to 160 brightness is a smaller gap than 0 to 255, but it also moves a lot slower for the first transition than for the second, and actually, the speed it moves is more important than the absolute distance. So monitors that are 25 ms on the ISO definition of response time (black-white-black) go up to like 50 or 75 ms in the gray transitions. However, manufacturers use "gray to gray" not "gray to gray to gray" (note that ISO means time to go black to white PLUS time to go white to black, the sum of two transitions, not one) so you should double any gray to gray numbers when comparing with standard black-white-black numbers (or, divide black-white-black numbers by two).
Overdrive just means to temporarily set the voltage to higher than what's intended. For example, if you're going from 32 to 160, you set it as if you want to go from 32 to 192 for one frame Then the next frame, you set it back to what you intended. So this way (assuming the screen reaches 75% of the difference per frame) at the end of the frame, it's already at 160 rather than 128. This takes advantage of the fact that larger transitions are quicker for LCDs than small ones. That's the idea, anyway. Manufacturers do their own thing with it. Note that the properties of the liquid crystal itself has not changed one bit -- this is purely the monitor's IC tricking the monitor into trying to produce another image, such that it becomes what you wanted by the next time. It is essentially a software trick rather than an actual hardware (i.e. LCD material properties) improvement.