• First, there are six center pieces, each of which has only one color or face. The centers can rotate freely, but never change positions relative to each other. In a standard cube, the rotation of the centers is not visible; some cubes are made with photos or designs instead of solid colors -- in these "logo" cubes, the rotation of the centers is important and requires additional steps to get right. We'll cover that later -- for now, use a plain cube with solid colors on each side.
• Second, there are twelve edge pieces, each of which has two colors or faces. Any edge piece may reside in one of twenty-four orientations (twelve distinct positions, and two orientations, or flips, in each position.) Because the centers never move relative to one another, the final position of any edge piece can be determined without regard for any other piece: an edge belongs in between the two centers that match the two colors on the edge.
• Third, there are eight corner pieces, each of which has three colors or faces. Any corner piece may reside in one of twenty-four orientations (eight distinct positions, and three orientations, or twists, in each position.) Because the centers never move relative to one another, the final position of any corner piece can be determined without regard for any other piece: a corner belongs in between the three centers that match the three colors on the edge.

• Top edges position and orientation

  The first step is to correctly position the four U/white edges (white-green, white-red, white-blue, and white-orange) so that they form a white-cross shape on the U/white side of the cube. When you are finished, the front and back of the cube should look like this:
  
  My algorithm doesn't give any explicit steps here; experiment until you can figure out how to get a white cross like this on your own. If you need help, there is an excellent description here.

• Middle edges position and orientation

  The next step is to get the four middle edges in their correct positions and orientations. These edges are the ones that have neither white nor yellow faces. Let's do the green-red edge first. There are four possibilities:
  • The green face of the green-red edge is on the D/yellow side of the cube.
    In this case, rotate the D/yellow side until its red edge is next to the red center:
    
    Then do the following sequence of four turns, RF'R'F:
    
    This four turn sequence has moved the green-red edge into its correct position and orientation without disturbing any of the other white or middle edges.

  • The red face of the green-red edge is on the D/yellow side of the cube.
    In this case, rotate the D/yellow side until its green edge is next to the green center:
    
    Then do the "mirror-image" four turn sequence F'RFR':
    
    This four turn sequence puts the green-red edge into its correct position and orientation without disturbing any of the other white or middle edges.

  • Neither face of the green-red edge is on the D/yellow side of the cube. For example:
    
    In this case, we will move an arbitrary (note 9) edge, shown in pink, from the D/yellow side of the cube into the spot occupied by the green-red edge, using the same F'RFR' sequence as above:
    
    This will leave the green-red edge with one of its faces (the red one in our example) on the D/yellow side of the cube, whereupon we can proceed as above.

  • The green-red edge happens to be in the correct position and orientation already.
    We got lucky, no need to do anything further with this edge.
  Repeat these steps for each of the three remaining middle edges (red-green, red-blue, and blue-orange). When you are finished, the front and back of the cube should look like this:
  

• Bottom edge orientation

  The next step is to get the four bottom edges in the correct yellow face down orientation. If we flip the whole cube upside down, so that the D/yellow yellow side is upwards and the L/orange side is to the right, the cube should (note 10) look like one of these four images. If two yellow faces are present, you may need to spin the down (D) yellow side to line up the yellow edges as shown.
  
  If it looks like the image on the left then you got lucky; proceed to the next step.
  If it looks like the second image, then you need to flip two of the edges without disturbing any other edge on the cube. Imagine a sequence of turns, S, that flips one of the edges we are interested in, while preserving the other three yellow edges. S is free to completely destroy all the rest of our progress thus far -- it only needs to preserve the edges on the D/yellow side of the cube. Then what you will do is execute SDS'D'. This will flip the one edge, totally mess up the rest of the cube, substitute a different edge, and then undo the mess while flipping the second edge back, and finally restore the D/yellow side to its original position.
  The whole sequence needed is FD'LDL'DF', which looks like this:
  
  
  The third image is very similar, except that you turn D twice in the middle: FD'LD2L'DF':
  
  
  The fourth image is handled the same as the second image, but repeated twice for each pair of edges that need to be flipped.

• Bottom edge position

  Turn the D/yellow side until the yellow-red edge is in the correct position and orientation. There are six possibilities for how the remaining three yellow edges can end up:
  • All four yellow edges are correct; proceed to the next step.
    
  • All three edges are incorrect; there are two ways this can happen:
    
    The first of these can be fixed using the following seven turn sequence, which rotates three of the bottom edges in a cycle, without affecting the other nine edges: L'DLD2L'DL
    
    
    The second of these can be transformed into the first using the same seven turn sequence: L'DLD2L'DL (note 11)
    
    
    Then proceed as above.

  • Two edges are incorrect; there are three ways this can happen:
    
    The first of these can be transformed into an analog of the first of the three-edges-different case by doing L and then rotating the whole cube 180 degrees to put the R/red side on your right:
    
    Then reposition the edges using the same seven turn sequence as before, just applied to different faces: R'DRD2R'DR (note 12)
    
    
    The second case is solved similarly, by doing L and then rotating the whole cube 90 degrees to put the B/blue side on your right:
    
    Then reposition the edges using the same seven turn sequence as before, just applied to different faces: B'DBD2B'DB
    
    
    The third case may be transformed into the first case by the same seven turn sequence: L'DLD2L'DL
    
    
    after which you proceed as above.

• Top corner positions

  Until this point, we have not paid any attention to the corner pieces. After this point, we will not pay any attention to the edge pieces: at the end of every following sequence all of the edges will remain solved, giving a familiar cross pattern on every side of the cube (note 13).
  Now rotate the whole cube to its normal U/white up, R/red right, F/green front orientation. We will begin by putting the white-green-red corner into its correct position (note 14). There are several possibilites to consider:
  • The white-green-red corner is between the D/yellow, F/green, and R/red sides with its white face on the R/red side:
    
    Do the following eight turn sequence, which cycles the DFR, UFR (shown in pink), and UBR (shown in cyan) corners in a cycle: R'D'RUR'DRU'
    
    
    
    Notice that this sequence has modified only three pieces: the entire rest of the cube (centers, edges, other corners) is preserved (note 15). This sequence will be used in one form or another to position all the remaining corners (note 16).

  • The white-green-red corner is between the D/yellow, F/green, and R/red sides with its white face on the F/green or D/yellow sides:
    
    Do the same sequence as above; this will place the white-green-red corner in the correct location, but in the incorrect orientation. Don't worry about that for now, we'll correct it later (note17).

  • The white-green-red corner is located in one of the other D/yellow side positions, for example:
    
    Here we do D' to bring the white-green-red corner into the same position as above:
    
    Then do R'D'RUR'DRU' as above, giving:
    
    Lastly, do a D to "undo" the D' we did originally and restore all the edges to their correct locations:
    
  • The white-green-red corner is between the U/white, F/green, and R/red sides, in any orientation. You got lucky; correct the orientation, if necessary, in the last step below. You can proceed to the next white corner.

  • The white-green-red corner is somewhere else on the U/white side. You got unlucky, but recall what happened to the pink and cyan corners above. Suppose that the white-green-red corner is where the cyan corner was in our previous example:
    
    We can do our same old R'D'RUR'DRU' to push the white-green-red corner down to the DFR corner where we'd like it to be:
    
    and then do another R'D'RUR'DRU' to get it where we want it:
    
    If the white-green-red corner is not where the cyan corner was, just turn U until it is, do the pair of R'D'RUR'DRU' sequences, and then turn U back so that the edges are correct again.
  Now rotate the cube 90 degrees so that the B/blue face is to the right:
  
  You can use the exact same sequence (transposed to B'D'BUB'DBU' to reflect that we're holding the cube in a different position) to put the white-red-blue corner in its correct location, giving (note 18):
  
  Then rotate the cube another 90 degrees so that the L/orange face is to the right:
  
  The same sequence (transposed to L'D'LUL'DLU') can be used to put the white-blue-orange corner in its correct location, giving (note 19):
  
  Then rotate a final 90 degrees, placing the F/green side one the right, to position the white-orange-green corner correctly:
  
  We can't just do a simple F'D'FUF'DFU', however, as that will move the already correct white-green-red corner that we started with. Suppose we have:
  
  Then before we use our F'D'FUF'DFU', we'll do an R, giving:
  
  Then we do the usual F'D'FUF'DFU', which gives:
  
  Then we undo our R with an R', giving (note 20):
  

• Bottom corner positions

  Rotate the whole cube so that the R/red face is up and the D/yellow face is to the right:
  
  We'll position the red-green-yellow corner first. There are only four places it can be
  
  The first case is easy -- it's in the correct position already, so proceed to the next step. The second and third cases can be solved by one or two repititions of our same three corner cycle, transposed to D'L'DRD'LDR' (note 21). The fourth case is a little tricky. First do D':
  
  Then do D'L'DRD'LDR' as usual:
  
  Then do D to put the edges back in place (note 22):
  
  Now rotate the whole cube so that the L/orange face is up and the D/yellow face is to the right:
  
  Now we simply repeat the same sequence (transposed to D'RDLD'RDL') zero, one, or two times (note 23) to get the remaining three corners in their correct locations, giving:
  
  If we're very lucky, all the corners will be twisted correctly, and the cube is now solved! But it's more likely that you'll have something like:
  
  in which case you should proceed to the seventh and final step.

• Corner orientation

  Just as we used a single sequence (R'D'RUR'DRU' and it's various transpositions) to position all eight corners correctly, we will use a single sequence to twist all eight corners correctly. This step is quite similar in spirit to bottom edge orientation above. Suppose that the cube looks like:
  
  We want to twist the white-green-red corner clockwise, and twist the white-red-blue corner counter-clockwise. Imagine a sequence of turns, S, that twists the white-green-red corner clockwise, while preserving the entire remainder of the white side of the cube. S is free to completely destroy all the rest of our progress thus far -- it only needs to preserve the U/white side of the cube. Then what you will do is execute SUS'U'. This will twist the white-green-red corner, substitute the white-red-blue corner for it, and then untwist the white-red-blue corner.
  S itself is rather long at seven turns: R'DRD'RDR, making the whole sequence sixteen turns long: R'DRD'R'DRUR'D'RDR'D'RU'
  
  
  Whew! That's S, now we do the key rotation of U to swing the white-red-blue corner into position:
  
  Then we do the seven turns of S': R'D'RDR'D'R:
  
  
  And finally, we do U' to put the edges back into place:
  
  We then repeat this sixteen turn sequence as many times as necessary, holding the cube in different orientations to select different pairs of corners, until all the corners are correctly twisted (note 24). There are several straightforward simplifications possible, for example you can twist the white-green-red corner clockwise and twist the white-blue-orange corner counter-clockwise by using SU2S'U2 instead of SUS'U. These are handy but not strictly necessary shortcuts, and are left as an exercise for the reader.

• Center rotation

  This step is only necessary when you are using a cube where the orientation of the centers is visible, for example when each side of the cube is a photograph instead of a single color.
  The principle is the same as for corners. We need a sequence of turns, S, that has some interesting effect on one or more center pieces, but does not have any effect on any of the edge or corner pieces, which we have already solved by this point. There are many such sequences, I will list two handy ones here. (No illustrations are provided, because the image generator I use only allows solid colors.)
  Option 1: Rotate the center of the U face counter 90 degrees counter-clockwise, rotate the center of the L face 90 degrees clockwise, and leave everything else unchanged. (This is sufficient to solve all the centers, but can be tedious.)
  Do: RL'F'BU'DLUD'FB'R'LU'
  Note that this is easier to execute mechanically if you think of it as moving the center slices of the cube, allowing the centers to change orientation relative to your hands throughout the sequence. So the first two turns are really "move the center vertical slice towards me by 90 degrees" and so forth.
  Option 2: Rotate the center of the U face 180 degrees and leave everything else unchanged.
  Do: URLUUR'L'URLUUR'L'
  This option isn't ever strictly necessary, but is easier and faster than using option 1 twice to achieve the same effect.
  Note that as with edge flipping and corner rotation, center rotation has parity. If you have solved the cube entirely except for a single center piece that is rotated by 90 degrees, then the cube has been incorrectly assembled and cannot be solved without disassembling it. This is not always obvious: consider that some cubes may contain some faces with solid colors and others with photographs -- you can always solve these even if they were incorrectly assembled, since you can use Option 1 above to fix the photographic centers while "hiding" any leftover parity violation in one of the non-photographic centers.

Notes