Q: How many squares are on an 8 x 8 checkerboard?  10 x 10? How many rectangles?

8 x 8 There are:
64 1 x 1
49 2 x 2
36 3 x 3
25 4 x 4
16 5 x 5
9  6 x 6
4 7 x 7
1 8 x 8 for a total of 204 squares

For 10 x 10
100 1 x 1
81 2 x 2
64 3 x 3
49 4 x 4
36 5 x 5
25 6 x 6
16 7 x 7
9 8 x 8
4 9 x 9
1 10 x 10

for a total of 385 squares

The rectangle counting is significantly more challenging.
Click here for a beautiful explanation from Nigel Coldwell