Beginner Method

Step 1. White Cross

Hold the white center piece on top, and find an edge in the bottom layer that has white on it. Note: An edge piece has 2 colors on it. Look at the edge piece's other color, and turn the bottom layer so the edge is under the center of the same color. Turn that face to bring the edge piece to the top.

Another example:

Anytime an edge piece is flipped (example above), fix it by doing the following moves:

Anytime you find a white edge piece that is not in the bottom layer, you can move it into the bottom by doing the following moves:

And then solve it like you would for any white edge in the bottom layer. Solve all 4 of the white edge pieces to make a cross. Make sure you always look at both colors on each piece so that you end up with the side colors matching as well.

Step 2. First Layer

Hold the white cross on the bottom.

Right handed 4-moves:

Left handed 4-moves:

Before solving anything, do this 4-move sequence repeatedly until you have it memorized. It will be very important later on!

Now we'll use the 4-moves to solve the first layer. With the cross on the bottom, find a corner piece in the top layer with white on it. Turn the top layer so the surrounding centers match the colors on the corner. Note: A corner piece has 3 colors on it.

Hold the cube so the corner piece is on the front/right, and repeat the right handed 4-moves until this corner is solved.

Repeat until all of the first layer corners are solved. If find a white corner incorrectly stuck in the bottom layer, you can bring it into the top layer by holding it on the front/right and doing the right 4-moves.

Step 3. Second Layer

Find an edge piece in the top layer without yellow as either of its 2 colors. Turn the top so that it matches a center. Face the piece and check if the top color matches the right center or left center.

If it matches the right side, do the following moves:

How to memorize this algorithm:
Turn the top with your right hand
Do the right 4-moves
Rotate the cube to face the right side
Do the left 4-moves

If it matches the left side, do the following moves:

How to memorize this algorithm:
Turn the top with your left hand
Do the left 4-moves
Rotate the cube to face the left side
Do the right 4-moves

Repeat this until all of the 2nd layer edges are solved. If an edge you are looking for is stuck somewhere in the 2nd layer, move any edge into its spot using one of the 2 algorithms above. This will cause the edge to come out into the top layer.

Step 4. Top Cross

Hold the cube to match one of the following (ignore the corner pieces)

Then do the following moves

How to memorize this algorithm:
Turn the front clockwise
Do the 4-move sequence
Turn the front counterclockwise
If the cross is not solved yet, hold the cube to match the new case and repeat.
Note: Focus on the colors on edge pieces, and not corner pieces. If you have 1 or 3 edge pieces facing up, your cube is unsolvable, and needs to be taken apart and reassembled.

Step 5. Match Cross Colors

Turn the top face until 2 cross pieces match the side color (if all 4 match, you have finished this step!)

Hold the 2 matching edges at the back/right. If they are across from each other, hold them in any way.

Then do the following moves

How to memorize this algorithm:
The first 3 moves of the 4-move sequence.
Look at the pair of pieces from the bottom.
Watch how they move away, and then back into the bottom.
Turn the top to match all 4 colors. If you can only match 2 colors, then repeat this step.

Step 6. Match Corners

A corner is in the correct position if all 3 colors on the piece match the surrounding colors.

Examples of corner pieces in the correct position

If 1 corner is correct, hold it in the front/right (if 0 are correct, hold any corner in the front/right).

Then do the following moves

How to memorize this algorithm:
Turn the top with your right hand, then turn the right side up
Turn the top with your left hand, then turn the left side up
Repeat, but go down instead of up
Check if all 4 corners are in the correct position. If not, hold a correct corner on the front/right and repeat.

Note: If you have only 2 corners in the correct position, your cube is unsolvable, and needs to be taken apart and reassembled.

Step 7. Solve the Cube!

It is very easy to make a mistake during this step, so its recommended to read the whole thing, including the common mistakes, before attempting it.

Turn the cube over so that the unsolved corners are all in the bottom layer.

You may have 2, 3, or 4 unsolved corner pieces.

Repeatedly do the 4-move sequence until the front/right corner is solved (has yellow on the bottom).

Then turn the bottom layer (not the whole cube) to bring an unsolved corner to the front/right. Repeat until the whole cube is solved.

Common mistakes:

Turning the whole cube to get the next corner to the bottom/right. Make sure you turn only the bottom layer.
Not finishing the 4-moves because the corner is solved after 3 moves. Make sure you always finish the 4-moves.

Note: If you only have 1 corner unsolved while the rest of the cube is solved, or if you followed step 7 correctly but it does not work, then your cube is unsolvable, and needs to be taken apart and reassembled.

Well done on solving your first Rubik's Cube!