Damage Over Time (DoT), also known as Periodic Damage, is a term used to describe skills or spells in Diablo III that continue to damage monsters even after the skill has been cast. Examples of such skills are the Witch Doctor's Grasp of the Dead and Haunt. Typically, a DoT skill is described as X damage over Y seconds. Some skills deal both instant damage and DoT.
Damage done by DoT effects is not applied at once, but drained gradually. Technically, target will suffer a hit 10 to 30 times per second (combined numbers popping up every 0.5 seconds). Some DoT effects can even roll a Critical Hit, or deal damage in a shorter duration (faster ticks) proportionally to Attack Speed, but most are simply increasing their plain tick damage proportionally to the Attack Speed, Critical Hit Chance and Critical Hit Damage. They will still be affected by the target's Armor and Resistances, if any.
Many DoT skills hit everything within the area of effect, effectively delivering a hit to every enemy standing in a 'pool', stopping the damage as soon as target leaves the dangerous area. Note that this damage counts as being inflicted in melee.
Channeled skills (like Disintegrate or Firebats) also deal damage like DoT, as long as a foe is foolish enough to remain in the area of effect of the beam or the torrent. These skills always roll normally for the Critical Hit and fully benefit from the Attack Speed, but drain Life from the target over time just like a normal DoT.
Unlike Diablo II, where heroes could not die from being poisoned, in Diablo III, both monsters and characters can die from any DoT effect. Note that most DoT effects can kill the target even after the player or monster who had placed that effect is already dead. Notable exceptions are Damage Over Time effects with very long duration — such as Firebird's Finery ignition or Haunt / Locust Swarm with Creeping Death.
Bleed is a special form of DoT, mostly using the Physical damage type, and is sometimes treated differently than other Damage Over Time effects.