Curses are black magic, no getting around it. They are any sort of malicious spell used on another.
Specific Details
Curses require great power, and are not treated lightly. A proper curse doesn't always die with the target - it can be passed on through the generations until it is broken or all those eligible for the curse die. All therianthropes can trace their shifting back to an initial curse, though they might be many generations removed.
A curse can be anything, from simple boils to an early, painful death for any offspring. They can be cast directly on the target, or via sympathetic magic. Alternatively, they can be cast onto an object, which then affects whoever uses or wears said object. In that case, destroying the object destroys the curse. They represent a concentration of the caster's malicious (or vengeful) intent.
Curses can sometimes be broken by meeting certain conditions, or lifted through very powerful white magic. They can also be lifted by the original caster (or their descendant).
Used By
Aes Sídhe
Powerful aes sídhe have been known to curse those who cross them.
Sorcerers
Sorcerers typically put curses on their things, so that no once else can use them. Most of these curses have short-term (often painful) effects.
Undines
Undines can curse those who offend them, rendering them incapable of breathing whilst asleep.
Witches
Most witches only curse if they have to, but once they do a curse lingers. Most witch-born curses have qualifications - they will be broken if and only if certain conditions are met.
Others
Gods (of certain polytheistic pantheons) have been known to curse those who displeased them, most notably the Gorgons.