by Victor Ortega and Josef Jelinek
For an introduction to the notation used in this page, click here.
This solution method is designed to solve Rubik's cube quickly, efficiently, and without having to memorize a lot of sequences or take a large business cash advance to pay someone to help you out. For ease and speed of execution, turns are restricted to the faces U, R, and F, and center and middle slices. Strong preference is given to R, since it is one of the easiest faces to turn for many people. Yet all sequences are minimal (or very close to minimal) by the slice metric.
This solution method orients cubies before positioning them. The idea is that it is easier to permute cubies after they've been oriented than before orienting them, because once the cubies have been oriented, the facelet colors that determine their permutation make easily identifiable patterns on the cube. Orienting cubies, whether done before or after positioning them, is always easy because orientation requires focusing on only one face color and on the patterns that that color makes on the cube. For middleslice edges on the last layer, permuting cubies after they've been oriented is a very simple affair, thus reinforcing this principle.
Do not worry about centers or edges while solving corners. Position centers while beginning to solve edges. You really only need to position U and D centers at that point, but positioning all centers may make things easier for you. Middleslice centers will be positioned along with middleslice edges on the last step.
This solution method is based on Minh Thai's Winning Solution. Ideas and sequences are borrowed from other solution methods, and appropriate attributions are made in those sections.
You should be able to manage this on your own. Don't worry about positionsall corners will be permuted in step 3. You should be able to orient U corners in 6 moves or less. For the greatest speed and efficiency, try to do this in one look.
Rotate the whole cube so that D becomes U. Orient the corners depending on which of the seven patterns below you see:
letter T pattern: R U R' U' F' U' F 

letter L pattern: F R' F' U' R' U R 

sune pattern #1: R U2 R' U' R U' R' 

sune pattern #2: R U R' U R U2 R' (inverse of #1 in both respects) 

letter pi pattern: R U R2 F' R2 U R' 

letter U pattern: R' F' U' F U R 

letter H pattern: R2 U2 R' U2 R2 
The ideas for this section come from this web page. Check it out for a different description of the process and to see several examples.
An "edge" here represents two adjacent corners on the U or D layer. Such an edge is considered to be solved correctly if the two corner cubies are positioned correctly relative to each other. A solved edge will be easy to identify because the two adjacent facelets on the side (not U or D) will be of the same color. A layer can have only zero, one, or four correct edges.
The number and location of correct edges can be quickly identified by merely looking at two adjacent side faces (that is, not U or D). For a given layer, if you see one correct edge and one incorrect edge, then there is only one correct edge on that layer. If you see two correct edges, then all four edges are correct. If you see no correct edges but both edges consist of opposite colors, then there are no correct edges on that layer. If you see no correct edges and only one edge consisting of opposite colors, then there is one correct edge on that layer, and it is opposite to the edge with the opposite colors.
Proceed with one of the following sequences depending on how many solved edges you have:
0: F2 R2 F2  
1 (DB solved): R U' F U2 F' U R' ((UB solved): R' U R' B2 R U' R)  
2 (DB and UB solved): R2 U F2 U2 R2 U R2  
4 (D solved): F2 U' R U' R' U F2 U R U R'  
5 (UF not solved): R U' R F2 R' U R F2 R2 
At this point, align corners and position centers. The cube is now fully symmetric except for edges. Pick the new U and D depending on what will make solving U and D edges easiest. Steps 4 and 5 can be combined, although this requires monitoring more cubies simultaneously and may not yield a speed gain or a reduction in number of movements. See this page for details on steps 4 through 6.
At this point, the last U or D edge will either be in the middle layer, in position but not oriented, or solved. Depending on the case, proceed as follows to solve that last edge (if necessary) while orienting the middle layer edges.
For another perspective on this process, check out Ron's approach
a) U edge in the middle layer
Position the "notch" at UR and the edge cubie at LF, with the facelet with the U color on the L face. If the edge cubie is twisted, mirror vertically (UR becomes DR, R becomes R' and R' becomes R)
In the diagram below, edges A, B, or C are oriented correctly (C) if that facelet's color matches the adjacent center or the opposite center. Otherwise it's incorrectly oriented (I).
(facelet D belongs in facelet position B) ("adjacent center" between A and C) 
Edge A  Edge B  Edge C  Pattern  Sequence 

C  C  C  R R' R' R  
C  C  I  R' R' R' R'  
C  I  C  R R2 R  
C  I  I  R' R R' R  
I  C  C  R2 R R R R'  
I  C  I  R' R' R' R R R  
I  I  C  R' R' R R  
I  I  I  R' R' R' R2 R 
b) U edge in position but twisted
There will be 1 or 3 twisted edges in the middle layer:
FR twisted: R U2 R' R2 R' U2 R' 

FR not twisted: R' R' R' R' 
c) U edge solved
There will be 0, 2, or 4 edges twisted in the middle layer:
2 adjacent (FL and FR): R2 F F' R2 F F' 

2 opposite (FL and BR): F F' R2 F F' R2 

all four: R F2 R2 R R F2 R2 R all four, when all edges are positioned correctly: F' U' F ( R)4 F' U F 
send FR to BL, BL to BR, and BR to FR: R2 R2 () 

exchange centers with opposites: () 

exchage FR with BR, FL with BL: R2 R2 () 