How many squares on the board

Count every square of every size on the board, not just the $64$ unit squares. A $k \times k$ square can sit in $(9-k)$ positions across and $(9-k)$ down, so there are $(9-k)^2$ of them. Summing over sizes $k = 1$ to $8$ is the same as adding $1^2 + 2^2 + \dots + 8^2$, which is also $\sum_{k=1}^{8} k^2 = \frac{n(n+1)(2n+1)}{6}$ at $n = 8$.

Organising the count in a table, one row per size, turns a confusing "count everything" into a tidy addition. Evaluate the sum yourself when a task asks for the total.