Everything but extra damage dice (flaming, sneak attack, vicious, etc) is multiplied on a critical hit.
Except instead of multiplying, you really reroll.
So, if Dumbo the Fighter with 18 strength and a +2 flaming greatsword scores a critical hit, he'll be doing:
2d6 + 4 (strength) + 2 (1.5x strength from two-handed weapon) + 2 (enhancement bonus) + 1d6 (flaming). Everything but the extra non-weapon damage dice is rolled twice for an effective damage roll of:
4d6+16+1d6
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Crits don't multiply damage, you roll it multiple times and add up the results. almost the same thing but it tends toward an average more
Weapon dice - Yes
Strength - Yes
Weapon Enhancement - Yes
Weapon Specialization (the feat) - Yes
Serrated Weapons (+1 bonus to damage) - Probably, don't know where this is from
Vital Strike (the feat and also it's improved parts) - NO, explicitly states in the feat "These extra weapon damage dice are not multiplied on a critical hit, but are added to the total."
Flaming/Frost/Shock (magical weapon ability) - NO, explicitly stated in the rules "additional damage dice from special weapon qualities (such as flaming) are not multiplied when you score a critical hit"
Bane (magical weapon ability) - NO, see above
Holy/Unholy/Axomiatic/Anarchic (magical weapon ability) - No, again
Merciful/Vicious (magical weapon ability) - NO, again
Sneak Attacks - Nope
Not sure if this is a definitive answer, but aside from any exceptions I might not know of a very easy rule of thumb is this:
Static number sources of damage are multiplied on a crit. Ex: (+2, +5, +20, ect)
Variable number sources of damage are not multiplied on a crit. Ex: (+1d6, +4d4, +2d10).
This rules out sneak attack and weapon properties like shock/flaming/frost.
The only variable that source of damage that I know that gets applied to a crit is weapon damage (rolled times the crit mod), or scoring a crit with a spell.
Static bonus damage tends to apply, but unless it's otherwise mentioned in the rules for whatever source of static damage you're applying, or if it's precision damage or something similar. Ex: Vital strike states in the feat description.
here is the complete text on critical hits, with the relevant sections bolded.
Critical Hits: When you make an attack roll and get a natural 20 (the d20 shows 20), you hit regardless of your target's Armor Class, and you have scored a “threat,” meaning the hit might be a critical hit (or “crit”). To find out if it's a critical hit, you immediately make an attempt to “confirm” the critical hit—another attack roll with all the same modifiers as the attack roll you just made. If the confirmation roll also results in a hit against the target's AC, your original hit is a critical hit. (The critical roll just needs to hit to give you a crit, it doesn't need to come up 20 again.) If the confirmation roll is a miss, then your hit is just a regular hit.
A critical hit means that you roll your damage more than once, with all your usual bonuses, and add the rolls together. Unless otherwise specified, the threat range for a critical hit on an attack roll is 20, and the multiplier is ×2.
Exception: Precision damage (such as from a rogue's sneak attack class feature) and additional damage dice from special weapon qualities (such as flaming) are not multiplied when you score a critical hit.
Increased Threat Range: Sometimes your threat range is greater than 20. That is, you can score a threat on a lower number. In such cases, a roll of lower than 20 is not an automatic hit. Any attack roll that doesn't result in a hit is not a threat.
Increased Critical Multiplier: Some weapons deal better than double damage on a critical hit (see Equipment).
Spells and Critical Hits: A spell that requires an attack roll can score a critical hit. A spell attack that requires no attack roll cannot score a critical hit. If a spell causes ability damage or drain (see Special Abilities), the damage or drain is doubled on a critical hit.
Well...they're wrong. It quite clearly says you roll your damage more than once. It doesn't say you roll your weapon's base damage dice more than once.
You just roll the damage, like you normally would on any other hit, more than one time. Excluding extra damage dice, of course.