Answer:
64 squares.
EDIT
There are 64 1x1 squares on a standard checkerboard, but there are also squares of other sizes.
There are;
64 1x1 squares
49 2x2 squares
36 3x3 squares
25 4x4 squares
16 5x5 squares
9 6x6 squares
4 7x7 squares
1 8x8 square
So in total there are 204 squares on a standard checkerboard.