Peter Renzland's Easy English Elegant Rubik's Cube Solution:

The Face Positions are: Up, Down, Back, Front, Left, Right.
"R" means "turn the Right face clockwise a quarter turn", etc.

1. Do the Up (white) Corners intuitively, properly aligned.
2. Place the Yellow Corners: Turn the yellow face so that at most 2 corners are displaced.
Hold them between the left thumb and index finger, at the Left-Down edge of the cube,
and swap them: URUR URUR URUR URU.
(If they are diagonal, make them adjacent for the swap.)

3. Orient the Yellow Corners (put them on the Back face): If 1 yellow corner cubie
is properly aligned, turn the cube so that it is Up Right BACK. Otherwise turn the cube
to place a cubie there, with its yellow facelet UP. (The White corners are always Front.)
Do: ULDR ULDR ULDR. (Repeat all of step 3 until the corners are done.)
4. Do the Edges: To move an edge cubie, align it opposite its home cubicle,
properly oriented (Up Front-Back); line up a helper edge-pair (already matched, or both still
random) alongside (Right Front-Back); Do: UURR UURR UURR; invert the line-up turns
(having noted their face-colours).

Intuition assistance for Step 1:

To move a White corner cubie from the Down face into its own cubicle:

Hold the cube so that the target cubicle is at Up-Right-Front.
Turn the Down face to move the cubie to Down-Right-Front.
    If the cubie's White facelet is Down, do   DDD RRR DD R B R BBB.
    If the cubie's White facelet is Right, do   LLL FFF L.
    If the cubie's White facelet is Front, do   B R BBB.
If the cubie is on the UP face, move it to the Down face first.

Intuition assistance for Step 4:

Q: What if 2 edges are flipped in place?
A: Swap them; then send them home properly oriented.

In general, to move a Cubie into a Cubicle, we need to realize that an Edge-Cubicle lies on
2 Faces, and therefore has 2 opposite Cubicles. We need to move the Cubie into one of the
2 opposite Cubicles, properly oriented, or else it will be flipped when we send it into its
target Cubicle. To move a Cubie into a specific Cubicle takes at most 2 turns. But to
move it there properly oriented may take 3 turns. To align a helper edge-pair may take
another turn. Then, holding the cube correctly, we apply the edge-pair swapping Transform
(RRUURRUURRUU), and then we undo the line-up turns, in reverse. When remembering
the line-up turns, it helps to refer to them by Colour, not by position.

A convenience Note on Step 3:

Do you find the sequence ULDR awkward?

Once you have learned this sequence, you may prefer this convenient way to apply it:
After "U", turn the whole cube (clockwise about the Front face) to put the Left face Up.
Then, instead of "L", you can simply do "U" again. And so on, and so on.

Keeping the centres correctly aligned may help align the corners and edges.

See also Peter Renzland's Elegant Rubik's Cube Solution Method.

[ peter [at] dancing [dot] org ]