First consider a 3 times 3 board:


The left figure shows that there are 9 squares of size 1 x 1. The middle figure show that there are 4 squares of size 2 x 2. The right figure shows that there is one square of size 3 x 3. Hence there are 9 + 4 + 1 = 14 different squares on a 3 times 3 board.

In the same way you can find the number of squares on a chess-board (8 x 8). Then there are 8*8 + 7*7 + 6*6 + 5*5 + 4*4 + 3*3 + 2*2 + 1*1 = 204 squares.

In the same way we can now solve the number of squares on a general n x n board. A similar calculation gives that the number of squares is in that case equal to: n * n + (n-1) * (n-1) + (n-2) * (n-2) + ... + 2 * 2 + 1 * 1 = 1/3 * n^3 + 1/2 * n^2 + 1/6 * n squares.