From D&D 3.5 to 5.0(5E) - 21 - Recovering Hit Dice - Different Than Hit Points

By now you probably know that when your character has a long rest they recover ALL of their hit points, even if you only had 1 at the end of the day and your hit point maximum is 100.  You might feel like Loki after confronting the Hulk at the end of the day, but your will feel 100% better the next morning.
You may also know that during the day you may spend some of your "Hit Dice" after a "short rest" to regain hit points.  This is a new use for the concept of "Hit Dice" that was not present in 3.5.
If my fighter is 8th level, then he has 8 Hit Dice and is an 8d10 creature.  Let's pretend the fighter has a 72HP maximum.  During the morning you fight off some small dragons and after the battle you are still down 35HP of damage.  You are on your own with no healing.  You can take a short rest, and then expend some of your "Hit Dice" to recover some hit points.

1. You decide to expend 3 of your "Hit Dice" to regain hit points so your roll a d10 three times.  (Fighters roll a d10 because they use d10 "Hit Dice".  If your character is a monk he would roll a d8 instead of a d10.  Check your class description for the Hit Dice roll to use.
2. As you roll the 3d10 you get a 7, a 4, and a 2 for a total of 13HP.
3. You are now down 22HP instead of 35HP.
4. But you would like to recover more hit points, so you tell the DM that you are going to spend 2 more of your 8 Hit Dice.
5. You roll a 7 and an 8 for a total of 15HP.
6. You are now down 7HP instead of 22HP.  Much better.
7. You have spent 5 of your 8 Hit Dice for the day.

The rest of your day is uneventful; you take your long rest and recover all of the hit points you were down.  You are now at your maximum of 72HP again.  But you have NOT recovered all of your Hit Dice.  You are starting out the new day with only 7 of you 8 Hit Dice available to you.  This is because the rules state, on page 171 of the PHB, that you recover just "half" of your Hit Dice each day at a maximum.  Since your maximum Hit Dice is 8, you can only recover 4 of your Hit Dice after each long rest.

I think this is to discourage players from using your Hit Dice too casually.  You want to put some thought into it each day if you expect the next day may also be challenging.