Easy Solution (Level 1)
Cube Home
There are seven steps you can learn with minimal memorization to solve the rubik's cube. They are listed below. Keep in mind that it is possible to "get lucky" and skip any of the seven steps (if they are already done). There are other methods that are much quicker than this, but they require more memorization. If you would like to learn to solve the rubik's cube quickly, look at the more advanced methods.
Step 1
Form a cross on one of the faces, like this:
Whatever color you choose for the cross will be your bottom color. I recommend using white, because the edge pieces you'll need will be easier to spot on the cube. First, find an edge piece with your bottom color, and put it so that both of its sides match up with their respective centers, like this:
Doing it three more times will yield this (notice the cross on the bottom):
If the cube looks like this, it is wrong:
You are now done with the cross.
Step 2
Finish the bottom by solving the four corners that have your bottom color on them. Check if the corner is oriented this way:
You should solve this case with U' F' U F. Do not memorize this. Understand it. The first move gets the corner out of the way, so you can do the second move without disturbing it. The third move connects the corner correctly to the edge piece it goes next to, and the final move puts everything back into place. The corner can be situated in other ways, also. Use the following moves to solve them (You should be able to figure them out on your own).
U R U' R'
Do something like R U' R' or F' U F to turn this case into one of the first two.
or 
do something like R U' R' or R U R' or F' U F or F' U' F to turn this case into one of the first three cases.
There are other possibilities for the location of the corner- these cases are only when the corner is above the place it needs to go. You should be able to apply what you have learned to figure them out on your own. You are now done with the bottom layer corners.
Step 3
You should now have one solid face, your bottom face, with a solid band running around it. Try not to think of it this way, think of it as having finished the bottom layer of the cube. You must now do the middle layer, but there are only four pieces in it, since centers don't move (with respect to each other) and don't count. Find an edge piece in the top layer that needs to go into the middle layer (It is possible, but unlikely, that there are none in the top layer. I will explain what to do in this situation later). An easy way of finding one is looking for an edge piece in the top layer that doesn't have the top layer color. Spin the U face until the edge is lined up with one of its centers. This means spin the U face until your edge is one of the two pictures below, and do the moves next to the corresponding picture.
U R U' R' U' F' U F
U' F' U F U R U' R'
If there are no edges that you need in the top layer, then find an edge you need that is in the middle layer. It might be in the wrong place, or in the right place but flipped incorrectly. Use one of the two sets of moves above to displace the piece you need, or, in other words, place a wrong edge in the middle layer. When you do this it moves the edge you need into the top layer, so you can put it in the right spot. You are now done with the first two layers, and only one remains.
Step 4
The goal of this step is to create a cross on the top face. It will be different than the cross we made on the bottom, in that it doesn't have to match up to the faces on the sides. In other words, it would be fine if, when done this step, your cube looked like this:
Now, look at the top face. If there is an "L" on it, make sure it is backwards, like this:
Now do F U R U' R' F' to make a cross. If you have, instead of an "L", any of the following, you should do F U R U' R' F' to them to make them turn into an "L".
When you are done this step, your cube should look like this:
Step 5
Now, make the cross line up on the sides, like you did with the bottom cross. By turning the U face, you should come to one of these possibilities:
The sequence R' U' R U' R' U2 R solves the third case. If you get the first or second case, do the sequence in order to turn it into a case which is farther to the right. Now, the only thing wrong with your cube is the top four corners (1 or more of which might already be correct).
Step 6
Now you should not turn the U face anymore, unless it is part of a sequence, because you will mess up the edges, but you can, and will, rotate the entire cube occasionally (looking at it from a different side) in order to make your cube look like one of the cases. You will now place the corners. It doesn't matter if they are twisted correctly. The sequence R' U L U' R U L' U' solves the last case. Do the sequence to any of the other cases to turn them into a case which is farther to the right.
The only thing left to do is to twist the corners correctly.
Step 7
This step is more confusing than most. You will solve the corners one by one, but the bottom two layers will get messed up and restored, and sometimes messed up and restored again. Regardless, they will be solved at the end.
You could have two, three, or four corners that are rotated incorrectly.
Put one incorrect corner in the top-front-right position. Do R' D' R D twice, and if the corner still isn't solved, do R' D' R D two more times. The bottom two layers are now messed up but don't worry. Move the U face until another messed up corner is in the top-front-right position. MOVE U TO DO THIS, NOT THE WHOLE CUBE! Again, do R' D' R D twice, and if the corner still isn't solved, do it two more times. Keep rotating U and solving corners until they are all done!