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Okay, I grew up playing with these things all of the time. The little toy that has like, Graham's number in different possible solutions, (519,024,039,293,878,272,000 to be exact) this class will focus on how to solve the damn thing. (And FYI, I have solved one before-my personal favorite-The classic 3x3x3. I cannot seem to find the 4x4x4 anywhere though.)
Okay, this is very tough for me to put in lamen's terms, since solving a Rubik's cube involves a lot of math formulae to understand, so you might learn something here.

Before we get started, we must work out a method for describing the various moves that will be made. There are six faces, with the following notations:

* Upper, or top face = U
* Down, or bottom face = D
* Left face = L
* Right face = R
* Front face = F
* Back face = B


We can turn each face either clockwise or counter-clockwise, with respect to the center (i.e. a move that may be clockwise to you, when looking at the cube, may not be clockwise for that face, in relation to the middle of the cube). The names for the different kind of moves (I'll use the U face as an example) are:

* A 90-degree turn clockwise on a face is denoted by U.
* A 90-degree turn counter-clockwise on a face is denoted by U' ("U prime") (Also note this is the same as U, done three times).
* A 180-degree turn either clockwise or counter-clockwise on a face, is denoted by U2 ("U squared") and is the same as two clockwise turns, or two counter-clockwise turns.

First, we put the edge pieces (those with 2 colors) in the right place. There are several possibilities. Note that when moving pieces around you should have the piece that you are working on the F face.

* White/Other color (OC) piece is on the D face. Rotate the D face so that it is directly underneath the place where it needs to go (on the U face). If the white side of the piece is on the D face, apply F2, and it will be correct. If the white side of the piece is not on the D face, apply D R F' R'.
* White/OC piece is on the center slice (i.e. middle portion of the cube) . Apply F or F' to get the piece on the D face, and then perform above moves.
* White/OC piece is in correct position, but incorrectly rotated. Apply F2 and do the above moves as necessary.

Anyone?

Okay, I'll simplify my teaching here.

Let's say that you're rubik's cube looked like this, the good old classic 3x3x3:

Rubik'sCube

The following seven pictures show how this cube is solved.
(Picture 4 und picture 6 may be different.)


Solution

We'll break it down now into smaller steps.

2nd step: Solving the corner cubelets.
There are four corner cubelets with the colour white. You hold the cube so that white is at the top and blue in front.
E.g. you have to move the corner cubelet white/red/blue to its correct position front/right/top.
First you move this cubelet to the place front/down/left.
There are three possibilities 2a), 2b) or 2c).
2a) white left

Example 03

2b) white in front

Example 04

2c) white underneath

Example 05

In the same way you deal with the three remaining white corner cubelets.
Result: The top slice is completely white. The colors of the edge cubelets match the center cubelets. (Picture 2 of 7)

Building the last Slice top
You turn the cube to solve the last slice (white underneath, yellow at the top).

4th step: Switch the position of two cubelets.
You turn one of the four edge cubelets yellow/blue, yellow/orange, yellow/green, yellow/red to the face with the correct color (if yellow is not at the top, don't worry!). If the remaining cubelets are not at the correct positions, you have to switch them over the left edge cubelet at the top. Probably you will have to repeat this sequence. (Picture 4 of 7)

Example 08

5th step: Flip the edge cubelets
Though all edge cubelets are in the correct position, they might be wrong. Yellow is not at the top.
You have to flip them. You hold the cube so that the edge cubelet is right/top/middle. Now there are eight moves.

Example 09

If the cube is colored after these 8 turns, go further. Turn the next edge cubelet to be flipped to the same place right/top by turning the whole slice (Don't turn the whole cube). Then there are the eight moves again.
Result: There is a yellow cross. You turn the top slice so that the edge cubelets are at the correct places. (Picture 4 of 7)

6th step: Switch the positions of three corner cubelets
First you move the four remaining corner cubelets to the proper positions. Their colors needn't be correct.-
If all four corner cubelets are in the proper position, then the following sequence is unnecessary.
If all the four corner cubelets are at a wrong place after aligning , then there are a further eight moves:

Example 10

If all the four corner cubelets are still in the wrong places, you have to repeat the eight moves. The front face must stay in front.
After these moves one corner cubelet is in a proper place. Turn the cube so that this corner cubelet is back/left. Now you repeat the eight moves.
Result: The four corner cubelets are in their correct places. (Picture 6 of 7)

7th step: Rotating the corner cubelets.
The colours of a corner cubelet have to turn.
You hold the cube so that a corner cubelet is top/right/front.

Example 11

If yellow is not at the top of this special corner cubelet, you repeat these eight moves.
Then you turn the top slice (not the whole cube), so that the next corner cubelet is top/right/front.
You repeat the 8 or 2x8 moves.
If all the four corner cubelets are rotated, yellow must be on the top. Now there is a last step. Turn the last slice. The six faces have one color. FINISHED! (Picture 7 of 7)

Example 12/Finished Product

Ah, what a refreshingly complex and well constructed class ^_^

Personally I'm looking forward to any future 4x4x4 ones