To disassemble the cube, turn the top clockwise a bit so that the top face is about at a 30° angle to the lower two faces. Then turn the left face down. Eventually, if done right, one of the edges will pop out. Pulling out the remaining pieces is trivial.

After all the pieces are off the cube, coat all of the black spots of the pieces with Q-tips or tissues. To save your precious Vaseline, there really is no need to coat the interiors or to coat with huge globs. Remember to coat the core of your cube as well. Put the cube back together solved and you're ready to go. It should be assembled solved because chances are you'll assemble the cube with an unsolvable configuration (Murphy's Law).

Move sequences for the Rubik's Cube are usually notated in this format.
F = Face (The face you're looking at)
R = Right
L = Left
U = Up (Top)
D = Down (Bottom)
B = Back

A move indicating turning the face you're looking at clockwise is denoted F. Turning the face counterclockwise is denoted F'. The same goes for the other faces -- e.g., U' means turn the top counterclockwise.

First off, thanks to Ben DeLoache for teaching me this method. This is a simple, relatively inefficient method (compared to the methods used by the world champions -- namely the Fridrich speed method). After much practice, you should be able to solve it in around 60 seconds, as opposed to the Fridrich method which can be executed in 17 seconds. The nice thing about this method, however, is that it can be learned relatively quickly and only requires the memorization of 4 move sequences, as opposed to the hundreds of sequences required by faster methods. This method solves the cube by layers, a visible indicator of progress. Some other methods solve by corners-first.

The first step in this method is to solve for the "top cross", shown in the figure below. In this case, White is used as the top color. I could tell you a few quick ways of moving pieces to the top, but it's better if you figure them out yourself.

N.B.: It is essential not only to get the pieces to the top, but to also have the pieces' colors match with the sides. For example, the white-blue piece should be on the top facing the blue center.

Top Cross

The Top, solved

Solving the middle is significantly easier. It requires the memorization of only 2 moves, and each is a horizontal reflection of the other so they're easy to memorize. First, look at the bottom for a piece that doesn't contain the bottom color [yellow] (if there are none I'll explain what to do later). We want to place the piece in the middle at its correct edge. In the first figure below, we want to move the red-blue piece to the edge of the red and blue faces. To do this, we look at the bottom color of the piece we wish to move (in this case, blue). We move it opposite to the blue face, as shown in the diagram.

Now we want to either move this piece to the right blue edge (in this case, red) or the left blue edge (in this case, orange). Since we wish to move the piece right, look at the blue face and execute the move-right sequence:

If there are no more pieces on the bottom to move into the center but the middle isn't solved, that means that the pieces are in the middle but not in the right spots (or not correctly oriented). Move the naughty pieces out of the edges by substituting a random yellow piece from the bottom, and then move the piece back in.

Moving a piece to the middle

After this, the top and middle layers will be complete. From here on, at the end of every move sequence the top and middle layers should stay intact.

The Middle, solved

Next, we want to obtain the bottom cross. This cannot be done the same way you did the top cross, because we must not disturb the top or the middle. Thankfully, there is a series of steps you can use that requires relatively little brain power. :^)

First, you're gonna have to learn what Ben calls "The Move" because it's used so often in solving the bottom. This is the third of four moves you have to learn. This is executing looking at the bottom face (yellow). The Move:

Pretty simple, eh? So the first step in finding the bottom cross is getting an "L" shape, shown in the figure below. For the rest of the method, you will be looking at the bottom (yellow). When I say "put the 'L' in x corner", that does not involve rotating a face. That means you rotate the whole cube (or your head) so that you're looking at it the right way.

Step 1: You don't have an "L". Execute "The Move" several times (it should only take one or two) until you get an "L". Look at the edge colors of the "L" (these are purpled out in the diagram because they could be anything but white). If they are opposites (Red-Orange or Green-Blue), then you're in luck -- goto 3. If not, goto 2

Step 2: You have an "L", but the colors are adjacent. Put the "L" in the bottom left corner and execute "The Move". If the colors are opposite now, goto 3. If not, goto 2

A Bottom "L"

Step 3: You have an "L", and the colors are opposite. Great. Now align the right leg of the "L" with its proper face color (rotate the bottom face this time, don't just rotate your head). This is shown in figure A below (it'll look like the left leg in the diagram because of the cube projection), and put the "L" in the bottom right.

Looking at the bottom, the right edge should be yellow-x, where x is the color on the right face, the bottom should yellow-x', the opposite of x (red ↔ orange, blue ↔ green). Look at the left edge. If that color is the same as the top face color, you're all set, goto 5. If it's yellow, randomly do the move and try again. If it's the same as the color on the bottom face, goto 4.

Step 4: You have an "L", but the left edge is the opposite color :^(. Orient the cube so the "L" is in the top left corner. Do the move, rotate the cube clockwise, do the move again, rotate the cube counterclockwise, do the move again, rotate the cube clockwise. You should end up with a correct "L". goto 3.

Figure A

Step 5: You have an "L", it's correct and in the correct orientation. Do the move. You will have the bottom cross. If you don't, goto jail. Do not pass Go, do not collect $200. You did something wrong, start the bottom over. If you have the bottom cross, it should look exactly like the figure below. Proceed to 6.

The Bottom Cross

Step 6: You have the bottom cross, and want to place the corners. Next step involves the memorization of the final move, we'll call it the corner move. What we want to achieve by doing this is to have all of the corner pieces in the correct spots, but not necessarily the correct orientation. So basically, e.g., the orange-green-yellow piece will be in the right corner, but the yellow might be facing the green face. That's okay.

The corner move (remember to look at the bottom face!):

Step 7: Your corners are in the correct spots. The best way to learn the next couple steps is by your own experimentation. What you want to do in this step is to get one corner piece correctly oriented, and three of them incorrect. All of them, however, should be in the correct spots. The move sequence you'll be doing is slightly tedious, we'll call it A: Do the move, rotate entire cube clockwise, do the move, clockwise, do the move, clockwise, do the move. Define B as A except rotating counterclockwise each time. We'll use B later. After you figure out how that changes the corners, figure a way to get the desired position. Hint: The top-right corner before A will remain the same. goto 8.

Step 8: One corner correct, three incorrect. This is the final step in solving the cube (phew!). Orient the correctly-aligned corner in the top-right (rotate cube, not face). Look at the top-left piece. If the yellow sticker is on the top, do A. If not, rotate the cube clockwise and do B. After this, your cube should look like the diagram below.

Solved Cube