The Basic Process  Stage 1
The first thing you need to do is subdivide the problem. You are not going to simply twist the cube a few times and find it solved (unless it was almost solved already). The way I originally learned to do this was by solving the top layer first. The following pictures will show you one example of this (the pictures show the exact same cube from 2 different vantage points):
Note that solving a top layer should be pretty easy, even for a beginner. Solve the edges first, and then the corners. Here are some routines you can use to help you solve the top layer:
If you are trying to solve the top layer first, the first thing you need to do is to decide what color you are going to have on top. When first starting out, it's easier to always have one particular color on top. This will make pattern recognition easier for you. The first thing you need to do is to get the edges into place. In other words, I usually get the whiteorange edge, the whitegreen edge, the whitered edge, and the whiteyellow edge in place. This is very simple to do.
Getting the edges in place can be done with very simple manipulations of the cube. Which move you are using depends on the situation you have. The first two examples below are quite simple. The third example is made a little more complex because we want to put the redwhite edge piece in the right place, but we already have all the other edge pieces in place, so we need to get the redwhite edge piece in place without upsetting the others.
When you are done, make sure that you have all the edges in the exact right place and rotation. Then you can work on the corners. In all of the example sequences below, you have certain pieces on the top surface, the white one, which are already solved, and you are trying to get the corner pieces into the right place.
Here are some examples of moves to use to get the corners in place, once the edges are all set.
There are more variants of these moves that you can learn with practice.
Simple Techniques for Solving a 2nd Layer
There are two moves that I am likely to use in solving the middle layer. They are the Right Handed Edge Mover and the Left Handed Edge Mover. To demo these moves, we will flip the cube, so our solved top layer is going to become the bottom layer. Then, these moves as shown here will help you move edges off the top layer into the middle layer in the correct orientation. Here they are:
Right Handed Edge Mover Basic
Use the mouse to grab the cube and rotate it. Three edge pieces have changed locations. Make sure you find them all. In the first animation, the pieces that have moved are the yellowwhite edge, the redwhite edge, and the yelloworange edge piece. Notice how two edges from the top layer have moved, and one from the middle layer (the yelloworange one) has moved.
Of particular interest is that the yellowwhite piece moved from the top layer to the middle layer. Taking advantage of this will be quite helpful in solving the middle layer.
Left Handed Edge Mover Basic
As with the Right Handed Edge Mover, 3 edge pieces have moved. Once again, use your mouse to rotate the cube and find them all. Using your real cube, make sure you do each of the above Basic Move Sequences 3 times in a row to see what it does. Done correctly, this should return you to a solved cube.
To get a feeling about how to use this we will provide an example of one of the edge movers (we will use the reverse of the left handed edge mover) in action. Notice that the piece that we are moving into place is on the back side, and it is the redgreen edge piece. Also, we have moved the completed layer to the bottom to fit in with the demonstration.
Building a Portfolio of Basic Move Sequences to Finish
The way to solve this final stage is by learning Basic Move Sequences. Make sure you learn these sequences using a completely solved cube. It will be too difficult to recover if you are working with a cube where you have solved 2 layers, and try to see what they do. Doing this with a solved cube is definitely the way to go. The Basic Move Sequences will give you 3 skills:
 Switching the position of 2 corners
 Rotate 2 corners, without changing anything else
 Moving and flipping the edges in groups of 3, without changing anything else
With these 3 skills, you can solve any cube. So let's dig in, and look at the details, but let's first start with a couple of important definitions:
Out of Postion vs. Rotated
To help you visualize a cube with pieces out of position, here are two pictures of a cube with corner pieces in the wrong position (both pictures show the same cube from different angles).
Notice how the redbluegreen corner piece is in the place which belongs to the redblueyellow piece and vice versa. Let's take this a bit further, and look at a couple of pictures where 3 edge pieces are out of position (both pictures show the same cube from different angles):
You can see that the cube is completely solved, except that there are 3 edge pieces out of place. Let's also take a look at a couple of examples of rotated pieces. Here is an example showing two rotated (aka flipped) corners (both pictures show the same cube from different angles):
If you look carefully, you will see that the orangeblueyellow piece sits in between the orange, blue, and yellow sides, but it's orientation is wrong (it is rotated). The same is true for the redblueyellow piece. Now lets look at a couple of rotated edge pieces (both pictures show the same cube from different angles):
You can see that there are two edge pieces that are in the right location, but once again the 2 sides of the edge do not match up with their corresponding center piece.
Now that we have defined these terms, let's move on to the next step!
Switching the position of 2 corners
Now we will show a method for causing two corners to swap position. In addition to swapping the position of two corners, this move will also cause movement of the edges on the last layer as well. That is why we do this as our first step on the last layer. Since we have not yet solved the edges, we don't care if they move.
When we begin working on the remaining edges, and on the orientation (or rotation) of the corners, we will use move sequences that do not affect any other pieces on the cube, other than the ones we are working on.
When you begin to work on the placement of the corners, you will either have 2, 3, or 4 corners out of place (or 0, of course, in which case you can skip this step). The best way to start is to take two of the corners that belong on one side that are next to each other (for example, the 2 red corners), and move them to the red side. If you have white on top, this will be the yellowredwhite corner and the greenredwhite corner.
Then you can see if the yellowredwhite corner is on the yellow side or the green side. If it's on the green side, then the two corners are swapped. Note that if the 2 red corners are out of place, try this same exercise with one of the other colors, yellow, green, or orange.
If 2 corners are out of place you will need to do this move one time to get those 2 corners in place. If 3 or 4 corners are out of place then you will need to use the move two different times to get everything into place.
Once you are done, the corners may be rotated so they don't look solved yet, but you will have them in the right location.
Lars Petrus offers a method he refers to as Niklas. Niklas will cause the 2 corners closest to you (the ones out of place in the picture above) to swap with each other, without changing the position of the two other corners. Here are the steps:
Once you have completed the move sequence, study the results. What do you notice about the 4 corners? The two in front have swapped positions and rotated. The two in back have stayed in position, and also have rotated. What about the edges? None of them have rotated, but they have all moved one position to the left. Note that if you repeat the Niklas sequence 4 times that the edges do move back to the correct position.
However, the corners are not yet right. Even though they are in the right position, they are rotated. If you keep repeating the Niklas move sequence, it will take you a total of 12 repetitions to get back to the solved position. There are many patterns that you may come to recognize, if you spend lots of long hours with the cube, and many other moves you can learn to take advantage of this, but for now, the Niklas gives you a way to swap the position of 2 corners.
Rotate 2 corners, without changing anything else
Once you have all the corners in the right location, you will find that there are 4 possible scenarios: 2, 3, or 4 corners rotated incorrectly, or all the corners are correct. Of course, if all the corners are correct, we don't need to worry about it. To make things a bit simpler, let's assume that 2 corners are correct, and 2 are wrong, as in the following picture:
To fix this, orient the cube as shown below, and follow the move sequence step by step.
Once you have done this, you will have flipped two corners without changing the location of any other piece. You should take a completely solved cube and do this sequence to see what it does. Once you have done it, repeat the exact same sequence on the same two corners and see what happens. Then do it a third time, and you will see that the two corners move back to their original position. If you start with a solved cube and do the complete move sequence 3 times, you will get back to the solved cube once again.
The beauty of this maneuver is that you can rotate the position of 2 corners without changing anything else.
Moving and rotating the edges in groups of 3, without changing anything else
Now everything should be done except a few edges. They can be oriented a lot of different ways. Once again, you should start with a solved cube to see what happens when you execute the new move sequence below. There are 3 Basic Move Sequences that will get you there.
We have already shown you the Left Handed Edge Mover and the Right Handed Edge Mover, but we need to show you a modified version of those, plus one new Basic Move Sequence. As before, repeat these move sequences 3 times to see what they do, and how they move and rotate the corners.
Note that the second cube in each example shows the same move sequence, but in the reverse direction.

Right Handed Edge Mover Complete
The difference between this move and the Right Handed Edge Mover Basic is that the 3 edges that we are moving are all on the top layer. Since at this stage we have the two other layers solved, we need such a move to be able to keep moving forward without undoing the work we have already done.

Left Handed Edge Mover Complete
This version of the Left Handed Edge Mover has also been modified to move three edges from the top layer only.

Two Handed Edge Mover
You will need to study these move sequences so you can learn which one to use in which situations. Make sure you understand the direction in which the pieces move, and the nature of how they rotate their orientation as well. With this in mind you will know which move sequence to use at which time.
Once you understand exactly what each of these sequences does, you will need a maximum of 2 of these move sequences to solve the cube when there are only edges out of place and/or rotated on the bottom layer. However, which move sequences you need will vary depending on the exact state your cube is in when you get to this stage.
Comments
Wow... Just wow...
By Kashy AliYour article offers many detailed steps, which makes it easier to follow through. Thank you!
By Sylvie LeochkoFinally I know how. Thank you.
By Jason KayGreat article, the illustrations really help a lot!
By Riley KleinThis is the first time for me with this issue, UNIQUE!!
By Elhusseiny ShahinGosh, who knew there was so much involved in solving a Rubik's cube!
By Anneliese BennettWow....that was fantastic... I love puzzles and mind stimulating games.
By Waheedullah AlekoThere's also a missing picture under the section "Rotate 2 corners, without changing anything else". The first paragraph refers to a picture to demonstrate 2 wrong coners and 2 right. But, there's not picture following the paragraph. Looks like it's missing.
By Charlie HubbardI see an error. In the section labeled Out of Postion vs. Rotated. The paragraphs above the pictures claim that it's the same cube from different angles. The last two sets of pictures in that section are mixed up. The right hand pictures should be swapped. Swap the right hand picture of the last set with the right hand picture directly above it.
So far that's the only error I see, and I'm on the final part! Thnx for the tut.
By Charlie HubbardCool text. Thanks! :)
By Łukasz Gręźlikowskiawesome!
By Tom Mungovan