Jon Tries to Explain How to Solve the Rubik's Cube
There are many many websites and videos explaining how
to solve the Rubik's Cube. Each does it a different way
and uses a different way to communicate. This is my
contribution which uses a combination of text, pictures,
mnemonic devices, and short videos.
Note that learning to solve the cube by watching videos is NOT cheating.
Only a persistent obsessive genius can solve it all by themselves.
Very few people can solve the cube  it's NOT easy!
If you learn to solve it you will have joined an elite group!
This is a beginner's method which is about the easiest to remember and do.
The aim here is to be able to solve the cube consistently and confidently.
If you want to go faster (which is a worthy goal but is certainly not necessary!)
you can consult the many resources at the end.
If you have a mathematical inclination you can see how
Group Theory applies to the cube. See the last section.
Cube Basics
The Rubik's cube appears to be 27 small cubes (cubies) arranged in a 3x3x3 stack.
It's actually something quite different and very clever.
There are 3 kinds of cubies:
 Center Cubies  6 of them  colored white, yellow, red, orange, blue, and green.
 Edge Cubies  12 of them  with two colors.
 Corner Cubies  8 of them  with three colors.
6 + 12 + 8 = 26. The 27th cubie is hidden inside at the center of the cube. You'll never see it.
The clever magic of the cube is that it can be twisted in every direction.
The videos on this site can be viewed in full screen. Choose to play it
here and then choose the little YouTube icon. Once on the youtube.com site
you can play it in full screen.
With just a few twists the cube becomes quite jumbled and scrambled up.
Your challenge  your mission (should you choose to accept it) 
is to restore the cube to the original state where
each face of the cube has same color on its 9 small faces.
Overall Plan
We'll solve the cube a layer at a time in 8 steps from scrambled to solved.
 Scrambled  The state your cube is probably in right now.
 White Edges  Matching the adjacent center cubie.
 White Corners  The first layer is done.
 Middle Layer  The cube is 2/3 solved! But we still have quite a ways to go...
The challenge is to put the last layer in order without disturbing the first two layers!
 Yellow Edges in Cross  Not necessarily matching the adjacent center cubie.
 Yellow Edges Properly Placed  Now they match.
 Yellow Corners Placed  Corners in right place but perhaps twisted.
 Yellow Corners Twisted = Solved!! Yay!
White Edges
The first step of getting the white edges in place is fairly easy and somewhat intuitive.
It does take some degree of 3dimensional aptitude.
This step is done by a series of simple short moves.
They are best explained with a video:
White Corners
Getting the white corners in their proper place is easier than the white edges.
There are essentially just two simple moves.
This completes the first layer. You should practice getting the first layer
in place several times before moving on to the next step.
You can solve the white layer, then the yellow, then the white, etc.
Middle Layer
This step uses just one fairly long move  with 8 twists. You can remember it with
the phrase "AwayDownBackUp". There is a symmetry to it that makes it memorable.
Again, practice this middle layer step several times before moving
on to the much more difficult 4 third layer steps.
Yellow Edges in Cross
The edges on the yellow face will now look like one of the following 4 images.
Ignore the corners  we're only interested in the edges.
A. 2 yellow edges across from each other.

B. 2 yellow edges in a corner shape.

C. No yellow edges.

D. 4 yellow edges.

Now it gets harder.
Our moves can't disturb the first two layers!
The following video introduces a notation
that is very helpful for specifying the moves and remembering them.
It explains the "FRUgal ruffle" move and how to use it.
FRONT RIGHT UP right up front = FRU ruf = FRUgal ruffle
With the image A. above you apply the FRUgal ruffle move once.
With B. you apply the FRugal ruffle move twice.
With C. you apply the FRugal ruffle move three times (thrice).
If you already have D. you are done with this step.
Note  Before doing the FRUgal ruffle moves make sure you reposition the cube
to match the images above.
What is a FRUgal ruffle? A decorative edge to fabric that is less expensive?
It doesn't really matter. It is a simply a mnemonic trick and it sounds fun.
Yellow Edges Properly Placed
Yellow Corners Placed
Yellow Corners Twisted
Alternate and Advanced Solving Methods and Other Resources
Group Theory
Group Theory is a branch of abstract mathematics that deals with symmetry.
It applies directly to the Rubik's Cube. It is not easy but if you have
the inclination and the time to study it can be quite beautiful.
See this introduction to group theory.