There are many different methods to solve a Rubik's Cube. Theoretically, the shortest solution is 22 moves from a given combination of colors on a mixed-up cube. However, no one has come up with a solution this short that works for all possible mixed-up cubes.

The method I'll describe here is easy to remember, but requires a pretty good imagination, since I can't use pictures or colors. It was derived from Matthew Monroe's presentation of the solution at:
http://www.unc.edu/~monroem/rubik.html
I find his directions to be easier to follow (since he has pretty pictures) but purely mechanical - he doesn't really explain how the moves affect the cube, he just tells you what to do to solve it. My directions attempt to tell you how each move changes the cube, so you can understand how the solution works, but this makes it lots harder to simply follow my directions and get a completed cube. So it goes.

Preliminary stuff
When I talk about the colors of my cube, I'm talking about the standard Rubik's Cube - some of them have different colors, and if someone's taken it apart the colors might be in different places. On mine, the colors are white, red, blue, orange, green, and yellow.

You have to realize that you can't really change the locations of the center pieces on each side. White is opposite yellow, blue is opposite green, and red is opposite orange. No matter how screwed up the cube gets, the center pieces of each side will always be like this. This is important - each face is defined by the center piece. If the center piece is green, that's the green side. If it's red, that's the red side. Your job is to move the edge pieces into place around those center pieces.

Look at one face of the cube. There are 9 ways you can move the cube - you can rotate one of the three vertical 'slices' (left side, vertical slice, right side), the three horizontal slices (top, horizontal slice, bottom), or the three stacked slices (front, middle slice, back). In these instructions, you will only need to move the front, left, top, and the horizontal and vertical slices. I'll write out moves in shorthand as follows:

<letter><number>
where <letter> is one of:
T, for Top
F, for Front
L, for Left
H, for Horizontal Slice
V, for Vertical Slice
and <number> is one of:
1 = "one quarter-turn clockwise"
-1 = "one quarter-turn counter-clockwise"
2 = "two quarter-turns clockwise", or "half a turn, either direction" since direction doesn't matter for a half-turn.

Examples:
L1 = "Turn left face 1 quarter-turn clockwise (towards you)"
T-1 F1 = "Turn top side 1 quarter-turn counterclockwise, then the front side 1 quarter-turn clockwise"

It's important not to change the orientation of the cube as you do these moves, because then you may forget which side is the left and which is the front and so on.

The solution, in 7 phases
I could have said "7 steps", but I dig the word phase.

Phase 1: Solve one side, with the edges correct (you should be able to do this yourself)
Phase 2: Position the corners of the opposite side
Phase 3: Orient the corners of the opposite side
Phase 4: Fill the edges of the opposite side (Keyhole phase)
Phase 5: Fill the keyhole
Phase 6: Position the 4 remaining edges
Phase 7: Orient the remaining edges (Rubik's Move)

RimRod says Thanks to your help, I was finally able to solve a goddamn Rubik's Cube. THANK YOU!
If RimRod can do it, SO CAN YOU!