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Please help a beginner to understand how a critical hit works.

From the Pathfinder Game Master’s Guide:

Critical Hit
Whenever a creature’s attack roll is a 20 on the d20, the attack might be a critical hit. Make another attack roll for the creature, just like the first one. If this second attack roll would hit the target, the original attack is a critical hit and does double damage—roll the damage for the weapon twice (including modifiers) and add the rolls together to find out how much damage you do. […]

To me this explanation sounds ambiguous, so please help me to understand this right. I'll give you an example fight. Let's assume a Goblin is fighting against Valeros the Fighter. Valeros hits first. The stats are as follows:

Goblin
Armor Class: 16

Valeros the Fighter
Weapon: Longsword
Attack Bonus: +4
Weapon Damage: 1d8+4

What I understand is how a critical hit may fail.

Failing Critical Hit
Valeros rolls a 20 on a d20, so his attack might be critical. He rolls again, but only a 11 on the d20. Even with his attack bonus of +4 his roll is lower than the Goblin's armor. The critical hit fails, but since the first roll was highter than the Goblin's armor, Valeros will deal 1d8+4 damage.

How much damage does Valeros do on a critical hit?

Valeros rolls a 20 on a d20, so his attack might be critical. He rolls again, this time 15 on the d20. With his attack bonus of +4 his roll is higher than the Goblin's armor.

The manual says that "the original attack is a critical hit and does double damage". At this point I haven't rolled for any damage. I would assume that the damage roll I do next will double. So 1d8+4 times two could be 2x(8+4)=24.

In the same sentence I read that I have to "roll the damage for the weapon twice". So I could also roll (8+4)(1+4)=17.

So what is true? How do I roll for damage?

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The thing I think you're missing here is that the description of "double damage" is all one sentence.

[...the attack] is a critical hit and does double damage—roll the damage for the weapon twice (including modifiers) and add the rolls together to find out how much damage you do.

Let's break it down.

the attack is a critical hit

Cool, now we've determined what it is, what does that mean?

and does double damage

That's a bit ambiguous. There's more than one way to think about "double damage" (as your post demonstrates.) But here comes the clarification:

roll the damage for the weapon twice (including modifiers) and add the rolls together to find out how much damage you do.

That's not a separate thing, that's the explanation of what they mean by "double damage" immediately before it. In your example, you'd roll (1d8+4)+(1d8+4), which might well be 1+4+8+4=17. Really, you're still doubling the damage you do, just before you roll instead of after.

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    \$\begingroup\$ That was needed. So double/triple/quadruple damage is really a stupid term if you didn't grow up with that. Thanks for the good explanation! \$\endgroup\$ – Daniel Sep 15 '16 at 0:01
  • \$\begingroup\$ I agree with this answer that the rules are clear, but personally my view is that the manual using the phrase "double damage" to describe this process is awful. "double damage" is entirely unambiguous as a phrase in and of itself--the fact that the manual insists on giving this phrase a new, wrong definition is terrible. It should say "and deals damage twice". Regardless of the thematic implications of this phrasing being "off", it would much more accurately represent the mechanical effect. \$\endgroup\$ – codetaku Feb 13 at 2:07
  • \$\begingroup\$ @codetaku But then it would set off all the "when a creature deals damage" effects twice. Really, it is doubling the damage - 4d6+4 is twice as much as (aka double) 2d6+2. \$\endgroup\$ – SirTechSpec Feb 13 at 3:14
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You roll twice the weapon damage and add twice other modifiers: (8+4)(1+4)=17

There is one exception, extra die other than the weapon base damage are not rolled again. For example, if you sword was flaming the extra 1d6 fire damage is not rolled again (in your example you would roll 2d8+8+1d6 for a +1 flaming longsword critical).

The exact way damage is multiplied is explained in the Damage section of the Combat chapter:

Multiplying Damage: Sometimes you multiply damage by some factor, such as on a critical hit. Roll the damage (with all modifiers) multiple times and total the results.

Note: When you multiply damage more than once, each multiplier works off the original, unmultiplied damage. So if you are asked to double the damage twice, the end result is three times the normal damage.

Exception: Extra damage dice over and above a weapon's normal damage are never multiplied.

Note there is another exception called out in the critical hit section, you never add more than once any precision damage:

Exception: Precision damage (such as from a rogue's sneak attack class feature) and additional damage dice from special weapon abilities (such as flaming) are not multiplied when you score a critical hit.

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A critical hit means that you roll your damage more than once, with all your usual bonuses, and add the rolls together. PFSRD Link

In your example you would effectively roll 2D8+8 which is double 1D8+4.

A critical hit may actually do more damage based on the critical hit multiplier of the weapon. Typically it is x2 (double) but some weapons have x3 (triple) or even x4 (quadruple) multipliers!

Some weapons also have a bigger critical threat range such as 19-20 or 18-20. Any natural attack roll within this range can threaten a critical hit.

Some types of damage are not multiplied by critical hits however such as precision damage or the Flaming weapon property. As a rule of thumb any dice that are not for the base damage of the weapon (e.g. d8 for a longsword) are not multiplied.

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