Damage
A damage roll is made to determine how much damage is dealt by a successful attack, area of effect spell, trap, or other damage source. It can involve one or many dice of any size from d4 to d12. It is separate to an Attack Roll, which is a 
d20 roll determining if an attack hits and thus whether a damage roll is made.
Damage rolls are expressed in the game using both dice notation (xdn, meaning x dice with n sides) and as a damage range ("(x~y)", meaning the roll will result in between x and y damage). All damage also has an assigned damage type, which interacts with creature resistances, vulnerabilities and immunities. If a weapon or other source of damage mixes different sizes of dice or damage types, they will be listed separately with a plus sign between them. The results of all individual dice are added together to determine the total damage (though see damage mechanics for more specifics on how this works).
For example, a successful attack with a 
Dagger does a base of 1d4
Piercing damage (1~4). This means a single four-sided die 
is rolled to determine the damage, for a total of 1 to 4 piercing damage. Most weapons use a single damage die, but some two-handed weapons use two: a successful attack with a 
Greatsword does 2d6
Slashing damage (2~12), rolling two six-sided dice 
for a total of 2 to 12 slashing damage. Damaging spells typically roll more dice: for example, being caught in a 
will cause 8d6
Fire damage (8~48), though a successful Saving Throw can reduce it to half.
Modifiers
For Weapon attacks, the attacking creature's Ability Score Modifier for Strength or Dexterity is added as a bonus to the total value of the damage roll. For the damage of Spell attacks, no such modifier exists, unless explicitly granted by a magical item, spell, or class feature (such as the Agonising Blast Eldritch Invocation for Warlocks).
The Proficiency Bonus and Advantage mechanics don't apply to Damage Rolls.
Ability Score Modifier
A bonus or penalty may be applied to the result of the roll based on either Strength or Dexterity. For rolls involving multiple dice, such as 2d4, the dice are rolled together, and the modifier is applied to the total result, not to each die.
Whether Strength or Dexterity is used depends on the weapon: usually Strength for melee weapons and Dexterity for ranged weapons. The exceptions to this rule are Finesse weapons, which automatically select Strength or Dexterity, whichever score is higher; and Thrown weapons, which use Strength for both melee and ranged attacks. If a weapon is both Thrown and Finesse, it uses the higher of Strength and Dexterity for both melee and ranged attacks.
Some examples, to make the possible combinations of Finesse and Thrown easier to understand:
- Using a Maul for a melee attack always uses Strength.
- Using a Rapier (Finesse) for a melee attack uses Strength or Dexterity; whichever the attacking creature has a higher score in.
- Shooting a Longbow for a ranged attack always uses Dexterity.
- Throwing a Handaxe (Thrown) for a ranged attack uses Strength.
- Throwing a Dagger (Finesse & Thrown) for a ranged attack uses Strength or Dexterity, whichever is higher.
Whether it's Strength or Dexterity that ends up being used, the following table defines the value of the modifier:
| Ability score | Modifier value | Ability score | Modifier value | |
|---|---|---|---|---|
| 1 | -5 | 16-17 | +3 | |
| 2-3 | -4 | 18-19 | +4 | |
| 4-5 | -3 | 20-21 | +5 | |
| 6-7 | -2 | 22-23 | +6 | |
| 8-9 | -1 | 24-25 | +7 | |
| 10-11 | +0 | 26-27 | +8 | |
| 12-13 | +1 | 28-29 | +9 | |
| 14-15 | +2 | 30 | +10 |
A bit of Mathematics
Note that due to the mathematics of dice rolls, the difference between, say, 1d8 and 2d4 is more than just the higher minimum value of 2 on the 2d4 roll. With the d8, you have an equal chance of getting, say, a 5 and an 8. On the other hand, the 2d4 roll is statistically more likely to lead to a total value of 5, than a total value of 8. This is most easily explained with a table of all possible outcomes:
| First roll | Second roll | Total value |
|---|---|---|
| 1 | 1 | 2 |
| 1 | 2 | 3 |
| 1 | 3 | 4 |
| 1 | 4 | 5 |
| 2 | 1 | 3 |
| 2 | 2 | 4 |
| 2 | 3 | 5 |
| 2 | 4 | 6 |
| 3 | 1 | 4 |
| 3 | 2 | 5 |
| 3 | 3 | 6 |
| 3 | 4 | 7 |
| 4 | 1 | 5 |
| 4 | 2 | 6 |
| 4 | 3 | 7 |
| 4 | 4 | 8 |
Notice how often the 5 appears in the possibilities for the total value (4 out of 16 possibilities) vs. how often the 8 appears (1 out of 16). This means a 2d4 roll has a 25% chance of resulting in 5 points of damage, but only a 6.125% chance of resulting in 8 points of damage. Meanwhile, the 1d8 roll actually has a higher chance of resulting in the maximum damage value of 8, since 1 out of 8 possibilities (12.5%) result in an 8.
