Hit dice

Hit dices are used to determine the amount of Hit points you have, they are also used as a limiting mechanism in spells. A PC gain an hit dice at each level, however the quantity of hit points given depends on the character's class. Monsters' hit dices are harder to determine as they do not have levels but challenge ratings (Monsters with the same challenge rating won't have the same number of hit dice).

Hit Points
The Hit dice is the kind of dice that is used to compute the number of hit points you gain at each level.

In DDO, there is actually no die roll: you always gain the maximum number of hit points. For instance, a Barbarian with a d12 hit die will gain 12 hit points at each level (plus its constitution modifier).

Spells
Hitdice refers to how many dice were rolled to generate your HP and it often a limiting mechanism in spells.

This is in order to keep spells with an hit dice limit like Sleep or Cause Fear as long as possible.