There are many ways to get really fast with the Rubik's Cube. I, myself, am not too fast compared to some, but I can explain how I have got to where I am at and how to progress further. With the method described, you can average under 14 seconds. The idea to get fast is to recognize, and execute. Basically, you are "learning" situations and knowing how to correct them.

First off, here are some common abbreviations:
F2L = First Two Layers
OLL = Oriente Last Layer
PLL = Permute Last Layer

The first two layers include the cross, each corner, and each edge above that corner (See first diagram in Step 3 of the Beginners Guide). The OLL and PLL parts are considered 1 "layer". Oriente is where you make the upper face color 1 solid color, in the method I use it is always yellow (Step 4 of the beginners guide). Permute is where you move the upper slice pieces around to their correct positions (Step 5).

Solving the cross is all intuitive. It is proven that the cross (the white + all the edges lined up w/ corresponding centers) can be solved in 8 moves or less no matter how scrambled the cube is. Over 50% of the time, the cross can be solved in 6 moves or less. See Dan Harris' page below for more information.

Probably the most common speedcubing method is the Fridrich method (see link below). What you do is solve the cross, insert the corner and the edge above that corner in pairs into the 4 slots, then oriente, and finally permute. Many of the top "cubers" in the world can solve the first two layers in 8-10 seconds, spend 2-3 seconds orienting, and 2-3 seconds permuting. The last layer (PLL/OLL) is just two simple algorithums, one for the OLL and one for PLL. Learning them is the easy part. Becoming very fast at executing them/recognizing which algorithm to use takes practice. Once the first two layers are solved, there are 57 different orientation cases (228 if you count learning from all angles), and 21 permutation cases (84 if you count from all angles). it isn't nescessary to learn every case to become quick. Many sub-20 average solvers only know a portion of the PLL and OLL algorithms. Getting the first two layers done quickly will improve your times greatly-however, it is the toughest part to master. The F2L consists of 42 "intuitive" algorithms. To become fast, you will want to be able to insert pairs into any slot from all angles. Learn to do this intuitively, watch the pieces move, and you will be able to learn to solve the F2L quicker. I have a couple speedcubing tips that have helped me out along the way:

• Practice mastering the cross. It took me around 3 months to master the cross. To practice the cross, try doing it with your eyes closed. Work on inspecting it for as long as you need to get it solved in under 9 moves, and eventually get down to where you only need to inspect it for 15 seconds or less.
  
• Practice, practice, practice. The more you practice, the more you will learn. The little things that help you speedcube come naturally from practice.
  
• Work on looking ahead. It is extremely important to be looking ahead at all times. While solving the cross, be looking for your first pair in the F2L, while inserting a pair in the F2L, be looking for your next pair, and so on. Start off going slow and looking ahead, and gradually increase your speed. There is no point in going quick on a pair without looking ahead and then having to stop to find another pair.
  
• When you are finished with the F2L, you have to oriente. Execute the oriente quickly, but not at 100% speed. While doing the OLL, be looking into the PLL. Once you get to the PLL, execute as fast as you can, because it is the last step of solving.