2x2 - Beginner's Method
Solve First Layer
Pick a colour and move corners so that the bottom face is completely solved.
Case 1
F' U' F
Case 2
R U R'
Case 3
U R2 U' R2
Last Layer Corner Orientation
Put the last layer corners in the right place by cycling three edges. Repeat as necessary.
Only Case
U R U' L' U R' U' L
Permutate Top Layer Corners
Apply the algorithm and repeat until a corner is correct. If there are more incorrect corners, move that corner into the top front right place and repeat.
Only Case
R' D' R D