One of the Necromancer's three skill trees, Curses debilitate enemies in various ways. The Necromancer's curses can do everything from blinding foes to having them attack each other and more, though none of them do direct damage. Unlike most skills, curses do not receive synergies.
While none of the curses directly damage enemies, they are all detrimental to enemies in some way. Some, such as Amplify Damage and Lower Resist, directly influence damage taken. Others, such as Confuse and Dim Vision, hinder their ability to attack the player. Different builds of necromancers benefit most from different curses; the lack of synergies allow for any necromancer to get the most out of their favorite curses without having to invest too many points in this tree.
Curses do not stack with the effect of Shrines that have lingering effects; activating one on a character removes the other. This makes certain shrines not very useful in areas where many monsters cast curse (such as the Chaos Sanctuary), but conversely make some shrines a handy means of removing curses.
In Diablo II, each target can only be affected by one Curse at a time (using another will override the previous one). In Diablo III, however, all three Curses can be active at the same time on a single enemy.