Helpless

A helpless character or creature is held, bound, sleeping, has any of its abilities reduced to zero, or otherwise completely at an opponent’s mercy.

Mechanic
A helpless character or creature cannot attack any target with melee, spells or any Spell-like ability (including a Beholder's eye rays).

A helpless target is treated as having a Dexterity of 0 (-5 modifier). Melee attacks against a helpless target get a +4 bonus (equivalent to attacking a prone target). Ranged attacks gets the same bonus. Rogues and characters with Backstabbing items can always sneak attack helpless targets if they are subject to sneak attacks regardless of agro.

Note that paralysis from a paralyzing weapon will prevent action but not cause this condition, contrary to one of the loading screens.

Helplessness is often caused by a spell such as Hold Person or Sleep. In the case of a Hold Person spell, casting Freedom of Movement will NOT end this condition, but will prevent future Hold Person spells while it lasts. To remove hold spells, use the spell Remove Paralysis. In the case of Sleep - you'll only get a single hit with the +50% damage, as attacking a sleeping target will wake it up.

If helplessness is caused by ability score attribute damage, it can be cured by any Restoration spell or Heal. Attribute damage also recovers with time.

Post-Update 9 Auto-Crit
"Auto-Crits" no longer exist. Helpless monsters now take an extra 50% damage from all sources - including sneak attack damage, spells, etc. Everything except glancing blows are increased by +50%. Attacks against helpless targets are no longer automatically critically hits. A critical hit can still occur if the attacker rolls in his threat range and will receive the same +50% damage. You can increase this rate up to additional 30% by Fury of the Wild's Sense Weakness.

Players that have been rendered helpless also take additional damage from all harmful sources, except for sneak attack damage and glancing blow damage, instead of being automatically critically hit by physical attacks, but the value varies depending on dungeon difficulty:
 * Solo/Casual: +5% damage
 * Normal: +10% damage
 * Hard: +15% damage
 * Elite: +20% damage
 * Epic: +25% damage

In addition to the extra damage, helpless creatures lose their dexterity bonus to AC and Reflex saves.

Coup De Grace
Coup De Grace is a related Pen and Paper DnD feature that does not directly exists in DDO. Instead for things which would quality for a Coup De Grace in Pen and Paper, qualify for helpless +50% damage bonus in DDO. Devpost on reasoning for this:

Note: Auto-crits were later replaced with the +50% damage bonus.