Step One
First make a central face. This can be done really quick.
Either find any two matching centers or pull any two together.
Then bring the other two matching centers to the face in front. When they are both there, twist them both up to the top.
Step Two
Finish the first layer. The top corners, one at a time, can be matched up with two corresponding centers and brought to the top layer to complete it.
So, pick a corner that matches the top layer colour and position it correctly underneath the top layer. If it is oriented wrong you can transit it round the cube to rotate it:
Before rotating it up to match the top layer, you must slot in the two matching center pieces on either side of the corner. This is fairly intuitive. Note that only one particular center will be able to go in to each place. When you have found one of the correct centers, move it until it is one twist away from matching with the corner. Now move the corner out the way, twist the center in, and return the corner. Do that for both centers and you'll have made a complete corner which you can twist up in to the top layer. Repeat for all four corners and you've solved half the puzzle.
Step Three
Now the other layer corners need to be sorted out. Time to flip the cube upside down and give them a good working over.
Orient the corners so they are the right way round. This can be done with a maximum of four twists on the bottom.
Once that is done, you'll need to permute them so that they are all in the right place. Here is an algorithm to swap the two edges closest to you, and an extension to swap two sets of edges.
Once the corners are done you should be left with only a few centers left to solve.
Step Four
This algorithm swaps the three marked centers in either a clockwise or anticlockwise fashion.
Clockwise:
Anticlockwise:
This algorithm is a bit more specific but is occasionally useful. It swaps the two cyans and the two magentas around.
It can be wrapped in to an algorithm which swaps a central half with an adjacent central half.
The End
If you spend enough time mucking around with the centers you should get this:
If the puzzle has been jumbled in to an unsolvable state, you should notice one of the following...
1) Two center pieces of the same colour can both get in to the same position on the cube.
2) You solve everything apart from two centers which are swapped.
If you are having jumble problems, check out my jumble guide which is a bit more in depth.