In order to solve a 2x2 Rubik's cube, we will be using techniques from the advanced 3x3 method. If you do not know the T-perm or Y-perm from 1 look PLL, and/or 2 look OLL, you should learn those first and then come back.
Step 1: Solving the First Layer
Solving the first layer is 100 percent intuitive. It is the same thing as solving the corners on the beginners method of a Rubik's cube, just you do not have the cross to guide you. You should be able to figure this out on your own. When you are done, the image to the left shows what it should look like.
Step 2: Orienting the Last Layer
Here is where knowing 2 look OLL comes into play. Perform the correct OLL to finish the 2nd step if you are getting confused on what OLL you have to do, imagine there is a solved cross separating all the corners. On the left is an example of the first two steps solved.
Step 3: Permuting the Last Layer
On a 2x2, only 3 permutations can occur. The 1st one is a solved cube. The 2nd one is a T-Perm, which occurs when 2 adjacent pieces need to be swapped. Hold the pieces that need to be swapped on the right and do the T-Perm to solve the cube. The 3rd one is a Y-perm. This occurs when 2 pieces are diagonally swapped. Do the Y-perm from any angle to solve the cube.