All you have to do is learn the patterns. The key to the patterns is that a completed pattern will only move 4 pieces, and will leave all the others as they are.
A Rubik's cube is solved in layers. You first pick a side (I usually start with White, since that's where the label is) and solve for that side, making sure all of the colors on the top layer match. Most people can do this part easily, but don't know where to go from there. At this point, one of the three layers is complete.
You next use two patterns to solve the middle layer, which really only consists of 4 pieces, as the center squares are all on a fixed axel. Now you have two layers complete.
Two more patterns are used to solve for the 4 corner pieces on the bottom layer. Finally, there are three patterns that are used to solve the 4 middle pieces on the bottom layer. This is the trickiest part, because you need to use 2 of the patterns in the right order at the right time for it to work correctly. It really just takes some practice and running the patterns over and over until you get it right.
The web links in the above posts demonstrate the patterns.
-Troy