A Cursed unit is very unfortunate. A very specific set of rolls are weighted against their favor; these aren't combat rolls, however, which are affected by Cursed Luck instead. A typical curse basically emphasizes Murphy's Law; if something can go wrong, it probably will.
A Curse is most noticeable through Afflictions. Whenever damage is taken, the chance that it will cause an Affliction is increased by 15%. When the damage is caused by Disease, the chance is increased to 50%. Both apply before reducing traits, however.
A Cursed unit with Horror Marks on them is especially unfortunate. Horror Marks notify the Horrors beyond the Veil of a potential meal, and Horror Marks on a Cursed unit are essentially counted twice.
The latter two effects (Affliction chance through Disease and Horror attacks through Horror Marks) were given by Cursed Items in Dominions 5, but this is no longer the case. This is a positive for Cursed items, a fair amount of whom are also Tainted and also apply Horror Marks.
A couple of items curse their bearers, often on top of being cursed themselves. Curses can also be applied by Cursed Magic Sites, such as The Shattered Throne. Most Curses will come from mages, however:
Certain units curse their attackers; Curse Bestowers do it all the time, while Death Curse bearers only do it to whoever killed them. Both are resisted by Magic Resistance, with their Penetration determined by the trait.
The only way to remove a Curse is through the Wish spell. Not even death removes it!