An attack that hits, a spell that lands, a fall that ends — all of them arrive here. Hit Points measure how hard a creature is to put down; damage takes them away and healing puts them back, and neither can push you past your maximum or below zero.
A damage roll is dice plus whatever modifiers apply, and it always carries a type from the list of damage types. The type has no rules of its own — its whole job is to be the thing other rules look at.
A Critical Hit is a natural 20 on the attack roll. It hits whatever the target's AC is, and it doubles the dice rather than the total: roll every damage die twice, add them together, then add the modifiers once.
Three of them, and each applies only once to an instance of damage:
A few tough creatures and objects also have a damage threshold, which is all-or-nothing rather than proportional: anything below the number is shrugged off entirely, and anything that meets it lands in full.
Losing half your hit points makes you Bloodied. That is a description other rules can refer to, not a penalty in itself.
At 0 hit points you drop, with the Unconscious condition, and you start dying.
At the start of each of your turns you make a Death Saving Throw: a bare d20 against DC 10, with no modifiers on it at all. Three successes and you are Stable — still unconscious, still on 0, but no longer rolling. Three failures and you are dead. A natural 20 puts you back on your feet with 1 hit point; a natural 1 counts as two failures. Regaining any hit points at all wipes the tally and ends it.
Damage taken while you are down is worse than damage taken standing up, which is why nothing about this is a stable position to be left in.
Healing of any kind restores hit points up to your maximum and no further; the overflow is simply lost.
Temporary Hit Points are the exception, because they are not hit points. They are a separate buffer that damage eats first, they do not stack with another lot — you take the better pool, not the sum — and healing never restores them.
Two rests do the ordinary work. Hit Point Dice are the pool you spend on a Short Rest to buy hit points back: roll one, add your Constitution modifier, and decide whether to spend another only after you have seen the result. A Long Rest skips the arithmetic — it returns you to your hit point maximum outright and gives back every Hit Die you spent getting there.