Other than removing the colored stickers and replacing them on the older models of Rubik's Cube, there is no one reliable trick that always solves a cube from any starting position. However, there are some relatively simple strategies that work given enough time.
A simple beginner's strategy devised by Ryan Heise is to first build a yellow cross on one side of the cube. Then rotate the cross until the colors near the sides start to match, and work the yellow corner pieces into the bottom layer. Then work in the four middle edges, so that two-thirds of the cube is properly aligned. This is followed by making the edges and the corners of the final layer face up and then correctly positioning them. This strategy may not be the fastest or most efficient way to solve a cube, but it works in time if applied properly and is designed to require the least amount of steps and memorization. Detailed pictures can be found at the creator's home page.
There are a number of other similar techniques that are more complex. Those who have gained some experience in solving the cube may want to look into "speed cubing" for more advanced methods.