Fridrich (CFOP) Method
The "CFOP" or "Fridrich Method" is currently the most popular speedcubing method. Most of the fastest cubers in the world (the ones that can average under 15 seconds) use this method. It was invented by Jessica Fridrich, and consists of four steps - Cross, F2L, OLL, and PLL. This method takes some time to learn because it consists of over 70 algorithms.
The first step of this method is to form a white cross (or whatever color you choose) on the bottom layer. Technically speaking, you solve the four first layer edges. Most people prefer to make a white cross because white stands out so this step and the following step are easier to do. This step is completely intuitive, and can always be done in 8 moves or less.
The second step is F2L ("First 2 Layers"). The first layer corner and the middle layer edge above it are inserted at the same time. This is usually done by "pairing up" the corner and edge in a way that they can be inserted in together. There are two main ways to pair the corner and edge. Like the cross, this step is intuitive, except for a few select cases.
Step 3: OLL
This is the third step of the Fridrich method and the first step of the last layer. In one algorithm, ALL of the pieces are oriented correctly, so that the entire top face is the same color. This is the hardest step to learn, consisting of 58 cases (including the solved one), and 57 algorithms to learn. It is usually learned last.
Step 4: PLL
This is the final step. Now that the top is all the same color (i.e. correctly oriented), the pieces must be moved around, o permuted, so the layer, and the entire cube, is solved. There are 22 cases (including the solved one), and 21 algorithms to learn.