Technically, there are an infinite number of possible amino acids, though we've found only 20 common ones that many organisms use and at least half a dozen obscure ones that specific organisms use. We focus only on the 20 common ones unless we're in a college level biochemistry course.
The three categories are nonpolar ones, polar ones, and charged ones. The nonpolar ones have non-polar side chains (no oxygens or nitrogens), the polar ones have polar side chains (most contain an OH group but no COOH group), the charged ones are either acidic or basic (has COOH group or a basic nitrogen).
