In the singular form, a hit die is the die rolled to generate hit points. In the plural form, hit dice (HD) is a measure of relative power that is synonymous with character level for the sake of spells, magic items, and magical effects that affect a certain number of hit dice of creatures.
Class hit die
A character rolls one hit die each time he or she gains a new level, then applies any Constitution modifier to the roll, and adds the result to his or her hit point total. Thus, a character has the same number of hit dice as levels. For his or her first hit die, a 1st-level character gets the maximum hit points rather than rolling (although Constitution modifiers, positive or negative, still apply).
Example
Vadania gets a d8 hit die because she’s a druid. At 1st level, she gets 8 hit points instead of rolling. Since she has a Constitution score of 13, she applies a +1 bonus, raising her hit points to 9. When she reaches 2nd level (and every level thereafter), Vadania’s player rolls a d8, adds 1 (for her Constitution bonus), and then adds the total to Vadania’s hit points.
Constitution penalty
If your character has a Constitution penalty and gets a result of 0 or lower after the penalty is applied to the hit die roll, ignore the roll and add 1 to your character’s hit point total anyway. It is not possible to lose hit points (or not receive any) when gaining a level, even for a character with a rotten Constitution score.