Curses are negative effects you can cast on enemies using gems with the Curse gem tag. They can reduce an enemy's damage, lower their resistance to certain types of damage, slow them down, or provide Life to units attacking them, among other things.
Curses, by default, are limited to 1 Curse per enemy. Various effects, including Ascendancies, items, a rare result of Corrupting an amulet, and one Passive Node near the top of the tree, can increase this limit. Such increases in limit are usually valuable, because Curses tend to be rather powerful effects. (You also technically have a limit of 1 Curse on you, but many hostile effects override this.)
Bosses are resistant to Curses (33% Less effect). Endgame bosses like the Shaper are even more resistant (66% Less effect). Rare monsters with the Hexproof modifier are entirely immune to Curses.