Beginner's Method

First, we will make a "daisy" by orienting all the white edges to the yellow side with white facing up. Once all the white edges are facing up, we can align each white edge to its other center piece and do an R2 move to place each edge on the white side. This creates a white cross on the bottom.

Next, we will learn a useful algorithm that will help us all the way to the end: [R U R′ U′]. You can apply this algorithm to many different situations. In this case, we will use it to complete the first layer.

Move one of the white corners over the space it is supposed to be, then, holding it on the right side, use the algorithm [R U R′ U′] to insert it into place. Repeat this algorithm until the white side of the corner is united with the white side; it may take up to five times. Rinse and repeat for the other three white corners.

Note: If one of the corners is already in place but is misoriented, simply use the algorithm until it is corrected.

Next, we will complete the second layer. Align an edge that belongs in the second layer on top of its appropriate center piece like the image.

(These next two algorithms include y rotations. For a y rotation rotate the cube so that the front side is on the left, but for a y′ rotation rotate so that the front side is on the right.)

Then, move the edge so that its top color is in line with its center piece and do the moves [L′ U L U y′ R U′ R′]. If the edge is the on the left side rather than the right like in the picture, do the moves [R U′ R′ U′ y L′ U L′]. Rinse and repeat for the other three edges.

Note: If the last two edges are in the second layer but not in the right places, insert a yellow edge into the second layer and then you can insert the second layer edge in the correct slot.

Now that we have completed the first two layers, we will get started on the last layer by making a yellow cross. If we only pay attention to the yellow edges, there will be only three cases to worry about, affectionately called the dot, the line, and the L.

For the dot, use the algorithm we learned at the start, with a little bit more: F [R U R′ U′] F′.
This will set up into the L case, where you will do a similar algorithm: f [R U R′ U′] f′. (Remember, a lowercase letter means to turn the first two layers.)
If you have the line case, you will do the same thing as the dot case: F [R U R′ U′] F′.

Next, we will finish the yellow face. Flip the cube over so that the yellow side is on the bottom.

To orient each corner, we will use the algorithm [R U R′ U′] until the bottom-right yellow corner's yellow side faces down. After the corner is oriented, use a D move to move on to the next corner. Repeat this until all the yellow pieces face in the same direction. It may look like you are breaking all that you just did, but do not worry, if you do [R U R′ U′] six times, the cube returns to its original position.

Now the yellow face is done. The sides will not be complete, but that is because we need to...

Permute the last corners! Flip the cube back over so the yellow side is on top.

As one of the last steps, we will learn an algorithm that permutes two corners and two edges. The algorithm [R U R′ U′ R′ F R2 U′ R′ U′ R U R′ F′] is called a T permutation. It may be a bit daunting at first, but it uses the same ideas as [R U R′ U′]. Just take it slow, you remember it in no time.

Note: If you need to swap two diagonal corners, use a T permutation to swap two corners, then do a y2 move and do another T permutation.

You're in the home stretch!

We will learn one final algorithm to align the final edges. This algorithm uses M moves, but it is not as hard as you may think. M moves can be executed by holding the left and right faces of the cube and using your middle or ring finger to move the middle column. M moves turn in the same direction as L moves—towards yourself—while M′ moves turn like R moves—away from yourself.

[M2 U′ M U2 M′ U′ M2] moves the left, front, and right edges in a clockwise direction. This is called a U permutation.

Note: If all four edges are misaligned, you will have to do this algorithm at least twice.

Congratulations! If everything went right, you solved a Rubik's Cube! You have bested all 43 quintillion different cases of a Rubik's Cube. Look at you go, galaxy brain! If you are especially proud of your time, why not submit your time?