Dividing the board into equal parts

Cut the $64$ squares into equal, congruent parts. For $k$ congruent regions each must have $64/k$ squares, so $k$ must divide $64$: the possible part-counts are the divisors $1, 2, 4, 8, 16, 32, 64$. Equal area alone does not guarantee congruence, so the shapes must also match under rotation or reflection.

Symmetry both generates and counts solutions: a division symmetric about the board's centre often splits into congruent pieces, and counting distinct divisions means counting up to the board's own symmetries so mirror images are not double-counted.