The only material you need for this instructable is your Rubik's Cube- so let's go! NOTE: You can view this instructable here, or download the abridged version as a one page .pdf
Use the picture as a guide- ULF would mean turn the Upper layer clockwise, then the Left layer clockwise, then the Front layer clockwise, whereas UL'F would mean turn the Upper layer clockwise, then the Left layer counter-clockwise, and the Front layer clockwise.
Step 1: The Cross
Now look at the edge pieces that make up your cross. Align all of them with the correct color of center piece that they touch. (This step is mostly intuitive, but might require flipping the F, R, L, or B layer twice to get the piece to B)
Step 2: Solving the Edges
From there, orient the cube with those pieces in F. Perform the algorithm L'UL to get the piece in place. Repeat this step only twice more with other pieces.
Step 3: White Corners
Step 4: The Final Edge
Step 5: The 2nd Cross
LUUL'U'LU'L'U' interchanges the edge pieces at UL and UB- repeat as necessary to align all the yellow edge pieces with their proper colors.
Step 6: Yellow on U
If your cube has either just a cross, or a fish pattern like picture one, find the yellow corner piece that when its side was rotated once, would end up in the empty space by it's tail. If, the yellow side of that piece is to the left of the center piece (as shown), orient it with that side to the front, and perform L'URU'LUR' -otherwise, (if the yellow side is to the right of the center piece) perform RU'L'UR'U'L. From there, reassess the pattern using this column.
If your cube looks like picture 2 or 3 (Double Fish/Castle), align it so the side with the arrow pointing to it (relative to the pattern- NOT the color). Then simply perform LFR'F'L'FRF' and re-evaluate the situation to find your next step.
Step 7: The Final Move!
Using this method, I've personally solved a cube in under 25 seconds on multiple occasions. Keep practicing- and don't be afraid to try new ideas! Once you get good, you'll want to work on 'finger tricks' -or using your fingers to spin the cube instead of your hands, in addition to keeping the cube oriented with one color on top and solving with any starting color!