How I solved Rubik's Cube
(this is not a tutorial)
I will show you here how I worked out my Rubik's Cube solution back in 1981. The purpose is not to teach you that solution but to show you the logic behind it.  Below is a step by step account of how I worked the puzzle out. I did go round in circles a few times so I have simplified it slightly and put it in a better order. In order to tell you a few moves I discovered I use the following notation- R = Right side clockwise move (looking directly at it) , R' is anti-clockwise move. The same applies to L for left, U for up/top, D for down/bottom and B for back. Please remember that I never read about solving puzzles though I have briefly seen some notation. You may therefore find some of my move descriptions unfamiliar. I will repeat that this is not a tutorial and it will only interest a small number of people. If you are looking for a solution I would suggest you work one out yourself as I have done since it is hugely satisfying. Once you learn someone else's it's very difficult to go back and devise a solution which is truly your own. There's nothing to stop you later learning another method for speedcubing.
My route to solving Rubik's Cube-

1. Complete top (red) side.

Approach-
Move red pieces into top side.

Problem with approach-
After two or three are in position inserting more removes existing ones.

Answer- Move the others out of the way before putting in new ones and then move them back. Other similar techniques become obvious.

Result-
One side now completed.

Problem-
One side complete but pieces in wrong positions.

Answer-
Start again but put red pieces in correct position as well as all in one side.

Result-
One side completed and correct.


2. Complete bottom corners.                                                                   

Approach-
Find moves that affect bottom corners yet leaves top red side untouched.

Problem with approach-
Only two parallel layers (middle and bottom) movable which don't destroy top red side. Not sufficient to change anything.

Answer-
Break up top red side very slightly, redo using different moves and see what affect it has on bottom corners.

Approach-
Take out one top red corner and put it back using different moves making note of bottom corners before and after move plus keep a record of the actual move itself. In my case I tried R', D', R, D', R', D2, R ( See video to right ).

Result-
One bottom corner untouched. Other three rotated. None moved. Using this move several times is enough to always orientate bottom corners. Very basic and time consuming but 50% of task 2 now completed. Improvements can come later.

2b. Find move that swaps corners only.                                                   

Approach-
Break up top red side again but slightly more. Each time write down bottom corner positions before and after move as well as move itself. Eventually through trial and error I found- R', D, L, D', R, D2, L', D, L, D2, L' ( See video to right ).

Result-
Two adjacent bottom corners are swapped. Orientation not affected. This was a bit lucky and meant I could now complete the bottom corners every time by first orientating them and then positioning. This takes less time than the other way round since it's easier to see what swaps are needed if corners are already orientated.
Top red side and bottom corners now complete.

3. Complete bottom side.

Approach-
Fill in bottom side edges.

Problem with approach-
Virtually no freedom to make moves without messing up solved parts of cube.

Answer-
Find way to free up areas of cube so edge manipulation is possible.

Approach-
Move a row of 3 top red cubies (corner, edge, corner) to bottom and under other reds- R2, D2. Frees up whole of R side as well as middle row between U and D sides. Huge amount of freedom gained through this simple move.
Now using these two 'free' layers manipulate pairs of edges into UR and DR positions. Nothing to write down or remember. Simple intuitive moves suffice ( See video to right ).

Problem with approach-
What if edges are in FD or BD positions?

Answer-
Move them to the free layers.


Approach-
For FD position- D, R, + middle and then R', D' to return to previous state. Now pair can be inserted as usual and red row can be returned to top. Repeat for other two bottom edges.

Result-
Top red and bottom side now completed.

4. Position final four middle edges.

Approach-
Use moves like R2, move middle layer and then R2 again. Fairly intuitive.

Result-
Edges now in correct positions.

5. Flip any incorrect middle edges. 

Approach-
Find very precise moves which only affect those parts (obviously). This bit is very difficult to explain. In order to understand it, lets consider a Rubik's Cube which is complete apart from 2 edges in positions RU and LU which need flipping. If you have absolutely no idea how to flip them what logical approach should you take? When presented with such problems I tend to do what I can and hope things will become clearer later. This actually works for a lot of things in life and there's no charge for that bit of advise.
So, What I know is that I could easily just flip the two edges though naturally that will mess up the rest of the cube. Lets start with flipping just one then but making a note of the moves and take it from there. It's not important how the edge is flipped but I do R, then use the middle layer to remove the edge by one 90deg turn. Then do R2 and use the middle layer to put the edge back in place but correctly orientated. Finishing up of course with R. So what have we got? Well, the top side is complete apart from the other edge at UL which still needs flipping and the rest of the cube is a total mess. We could of course reverse the move but that would unflip the edge we just flipped and get us no where. Hang on a minute! flipping an edge or unflipping it is surely the same thing in terms of what happens to the actual cubie. So, what if we did U2 and then reversed that set of moves. Wouldn't this unscramble the cube and unflip/flip the other edge piece? Yes it does. That difficult to explain bit of logic is one of the most important things I worked out about solving puzzles in general. (See video of this move to right)

Result-
Cube solved! All possible scrambles can now be dealt with.

On the right you can see a typical solve of mine using this method. My best time is around 28 seconds back in 1981. I never use inspection time since to me that's part of the solve and there for wrong. I believe it is a great shame that the official rules allow this 'cheat'.  
I haven't really bothered improving much of my solution but one thing I changed was corner orientating. Sometimes I'll stay with my original move but if just two need orientating I will use the same logic as my edge flip move.
I have often wondered if any of my moves are used by others but it is a principle of mine never to read solutions. This is not through any kind of arrogance, I just don't want to inadvertently learn new moves since I want to solve all puzzles 100% by myself. This approach may change one day though since I'd love to learn a really fast method for Rubik's Cube.
Since 1981 I have adapted my method to solve the 4x4x4, 5x5x5, 6x6x6 and 7x7x7. Out of those I have videos for my 4x4x4, 5x5x5 and 7x7x7 solve also.