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F2L

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 F2L stands for first two layers.  The F2L can be solved numerous ways, however the most common way is Fridrich F2L, which is to first solve the cross and then solve the four corner-edge pairs around it.  It is not necessarily the most efficient way of solving the F2L, but in my opinion it is the fastest.

Intuitive F2L vs. Alg-based F2L

 Usually, people will say to learn F2L completely intuitively.  That means you don't learn any algs, you simply figure out how the cases work for yourself.  I have always been against this approach for one reason:  move count.  Improving your F2L has a lot to do with reducing your move count, and I have found that people who learn F2L intuitively more often that not will waste a lot of moves.  For this reason, I recommend learning F2L by learning a list of algorithms.  By the time you get to the end of the algs, you should be able to understand how they work anyway.  So this is kind of a combination of the two approaches, with the emphasis on learning the algorithms.  With that being said, here are the algorithms in the order I recommend learning them.  Note that any time you see a cube rotation in the middle of the alg, you can use a d turn instead.  I personally dislike d turns, so I tend to just rotate the cube, though I have recently been trying to incorporate more d turns into my solving.  Try both options out and stick with the one you prefer.

U R U' R'

y' U' R' U R

R U R'

y' R' U' R

R U2 R' U R U2 R' d R' U' R

R2 U2 R' U' R U' R' U2 R'

R U R' U2 R U' R' U R U R'

R U' R y U R U' R' F2

R2 y R U R' U' y' R' U R'

(R U R' U')3x y' (U' R' U R)3x

U' R U' R' U2 R U' R' y' U R' U2 R U' R' U' R

U' R U2 R' U R U R' y' U R' U R U2 R' U R

R U' R' U y' R' U R y' R' U R U' y R U' R'

U' R U R' U y' R' U' R

y' U R' U' R U' y R U R'

U R U' R' y' U' R' U R

y' U' R' U R y U R U' R'

R U' R' U R U' R'

y' R' U R U' R' U R

R U R' U' R U R'

y' R' U' R U R' U' R

U' R U2 R' U y' R' U' R

y' U R' U2 R U' y R U R'

U' R U' R' U R U R'

y' U R' U R U' R' U' R

R U' R' U2 y' R' U' R

y' R' U R U2 y R U R'

R U2 R' U' R U R'

y' R' U2 R U R' U' R

y' U2 R2 U2 R U R' U R2 y' R' U R U R' U' R U2 R' U' R

U2 R2 U2 R' U' R U' R2 R U R' U2 R U R' U' R U R'

y' U R' U2 R U2 R' U R

R U2 R' U2 R U' R'

y' U R' U' R U2 R' U R

U' R U R' U' R U2 R'

U' R U R' U R U R' y' U R' U R U' y R U R'

U' R U' R' U y' R' U' R y' U R' U' R U' R' U' R

R U2 R' U R U' R'

y' R' U2 R U' R' U R

R U R' U R U' R'

y' R' U' R U' R' U R

How to improve your F2L:

-Always be looking ahead
-There's really no way I can explain how to do this, it just comes with a lot of practice.  But you should find a solving rhythm that you are comfortable with.  A good way to practice this is to solve with a metronome.  Set it to a certain tempo and do one or two moves for every beat.  If you miss a beat, start over.  There shouldn't be any pauses in your F2L; it should theoretically look like one big fingertrick.  Remember, it's better to slow down slightly and have no pauses than it is to go full speed with breaks in between each pair.

-Be able to perform cases from multiple angles
-You shouldn't have to learn any new algs to be able to do this.  This is where the intuitive part of F2L comes into use.  Consider the following cases:

L' U L U2 y' R U R'

L U L' U L U' L'

L U' L' U L U L'

Note that you are using the exact same moves as the normal algorithms, just from a different angle.  If you understand how the F2L algs work, you should have no problem doing this for all the cases.  This is a great way to work on your left hand dexterity as well.

-Use empty slots
-Often times, you can use empty slots to your advantage to reduce move count and avoid cube rotations.  Compare the following cases with the normal algorithms:

R U' R' U R' U' R

U' R U2 R2 U' R

R' U R2 U R'

These should be fairly straight forward to understand.  Again, this is another area in which you can use intuitive F2L to your advantage.  You can take the ideas presented here and apply them to plenty of other cases.

-Know tricks for special cases
-The following algs are very useful to know:

U R' F R F' R' U' R

d2 R2 u R2 u' R2

U R U' R' L' U' L

U' R U R2 U' R

R U' R2 U R

There are several of other cases like these.  So if you run into a tricky case, experiment a little and try to come up with algs of your own!

-Use working corner when you can
-Using working corner is a great way to cut down both your solving time and your move count.  Here are some examples of it:

D R U R' D'

U D' R U' R' D

D2 R U R' D2

You can take the same idea and apply it to edges:

D' R U' R' D

Sometimes you can make easy extended crosses using working corner techniques.  Be sure to take advantage of these easy cases when they come up!