Hit dice

Hit dice are used to determine the amount of hit points you have, they are also used as a limiting mechanism for spells. A Player Character gets 1 Hit Dice per character level, making the Hit Dice of a character equal to their character level, however the size of the hit dice and the number of hit points given depends on the Player Character's class. Monsters in Dungeons and Dragons Online have standard hit dice for their creature type, see Monster Manual.

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

Dungeons and Dragons Online follows the maximum hit point rule: you always gain the maximum number of hit points at each level. For instance, a Barbarian with a d12 hit die will gain 12 hit points at each level (plus its constitution modifier).

Spells
Hit dice refer to how many dice were used to generate your HP and it is often a limiting mechanism for spells.

This was done in order to keep spells with a hit dice limit like Sleep or Cause Fear useful for as long as possible. Sadly, it makes spells with hit point limits like the power word spells less useful.