This method is developed from my old beginner method, which in turn was basically copied from here, and turned upsidedown for easier progressing to more advanced methods.
This method has been composed with advancing to more efficient methods in mind. As such, not all the algorithms are the easiest possible, since some of the old beginners' algorithms are not usable with more advanced methods. The old method is described here for historical reasons, but if you plan to learn more advanced methods, start with the following method:
The first stage in solving the cube is solving the white cross. The end result should look like this:
All that's done here is that the edge cubies that have white in them have been moved next to the white center cubie. Note that also the nonwhite sides of the edge cubies must match the nonwhite center cubie they are next to! There are no algorithms for the white cross, you'll just have to figure it out.
After this, turn the cube upside down. The algorithms and the rest of the images are based on the yellow face being up. Or you can build the cross with the white face in the bottom to start with; this will make progressing easier later.
More indepth information about solving the cross can be found here.
Next we drop the first level corner cubies in place. This requires 3 algorithms, 2 of which are mirrors of each other. The mirror algorithms are so simple they're not really algorithms at all.
Start by moving the corner cubie you want to put in the right place directly above the right place, like this:
The algorithm you need to do depends on which way the white face of the corner cubie is facing:
White is on..  Apply algorithm: 

U  R2 U R2 U' R2 
F (like the example image)  F' U' F 
R  R U R' 
Repeat for all four corners.
After the previous stage, you should have something that looks like this:
Now, find an edge cubie that needs to be in the second layer, but is currently in the last layer. Move it so that the colour on the side of the cubie matches the R or F face center cubie. After that you can one of the following algorithms:
U R U' R' 

U' F' U F 
Note that the start and end halves of the algorithms are identical, just in different order.
If you end up with a situation where the cubie you need is already in the second layer, but in the wrong place or incorrectly aligned, use one of the above algorithms to move just any cubie in its place. This will lift your cubie into the last layer, and you can deal with it with one of the two algorithms.
Create a yellow cross on the last layer. The colours on the sides of the edge cubies in the cross don't matter at this point. There are four possible patterns that you may see on the last layer before doing the cross (ignoring the corner cubies for now):
If you see this..  ..do this: 

F U R U' R' F' 

F R U R' U' F' 

Do either of the above algorithms, and you should end up with a situation that can be solved with one of them. 
The fourth pattern is the one where the yellow cross is already finished, in which case you can move forward.
There are seven possible unsolved states at this stage. Any of these states can be solved by using only two algorithms that may need to be repeated. (actually, one would be enough; you'd just need to apply it more times). The algorithms are used to rotate three corner cubies CW or CCW:
Rotate orientation of cubies (apart from the Back Right corner) CW: L U L' U L U2 L' U2 

Rotate orientation of cubies (apart from the Back Left corner) CCW: R' U' R U' R' U2 R U2 
If you're cube is not in a state that can be solved by rotating three corners in the same direction, you'll just need to apply one of the above algorithms to get the cube in such a state. From any of the seven states, you need to apply a maximum of two algorithms to get the corners correctly oriented.
I'm not going to go into great depth explaining how it works; you should be able to figure it out. If you're having trouble, see Jasmine's beginner method page, or even the Further tips for LL corner orientation page.
Once the corners are oriented, you should have something that looks like this:
Next, we move the last layer corner cubies in their correct places, while preserving their orientation. There are two possible states here; The positions of three corner cubies need to be rotated, or two diagonally opposite corner cubies need to be swapped.
Before performing the algorithm, rotate the Ulayer so that the "head lights" (two corner cubies with identical colours on a side face, blue in this example) are on the Bface. If you don't have the "head lights", you need to perform the algorithm once to get them to show up.
Start this algorithm with the "head lights" on the Bface. If the "head lights" are not there, you can start from any position. Cycle cubie positions (apart from the Front Left corner) CW: R' F R' B2 R F' R' B2 R2 
Once the last layer corners are correctly oriented, the only thing remaining is to position the last layer edges to their correct places. If you got lucky, this stage is already solved, but if not, there are two algorithms that you can use to solve the cube:
R2 U' F B' R2 F' B U' R2 

R2 U F B' R2 F' B U R2 

Do one of the algorithms above, and you'll end up with a cube that can be solved with one of them.  
Here also; do one of the algorithms above, and you'll end up with a cube that can be solved with one of them. 
Your cube should now be solved; Congratulations!
Once you've learned the beginner method, you can continue to the intermediate method to learn a few extra algorithms to solve some of the stages in fewer moves.