A standard chessboard has 64 squares arranged in an 8-by-8 grid (eight ranks and eight files), with 32 light and 32 dark squares in a checkered pattern, according to the definition of a chessboard (Wikipedia, Chessboard). That is the count used in play: each piece stands on one of those 64 unit squares.
A common puzzle asks how many squares of every size you can find on the same board. Counting 1×1 through 8×8 squares gives 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 204 (Math is Fun, squares on a chess board). So the everyday answer for chess is 64; the all-sizes puzzle answer is 204.
Source: Wikipedia, “Chessboard”.
Image: Chessboard diagram, Encyclopedia Britannica (1911), public domain, via Wikimedia Commons.