Guide not completely updated for Wrath of the Lich King
In the game, when you interact or attack a mob, there's a hidden mechanic calculating and working to determine how a mob will react. This mechanic is referred to as threat, because as you attack the mob, eventually you will cause the mob to attack you; the mob attacks you because you were the most threatening to it. Understanding how threat works will help you to prevent pulling enough threat so that you aren't in danger, and you may increase your damage output by timing abilities accordingly.
Threat is a different thing than aggro: We define threat as the numerical value that the mob is accounting for to see how it should determine its attacks; aggro is whom the mob is actually attacking or focusing on.
Every mob has a list that keeps an account of player's threat values. If a player has not affected a mob in any way, then that player has not caused any threat. Mobs will react to this threat list, and determine who they attack or direct their attacks towards. You can gain threat on a mob's threat list by doing things, such as:
Body pulling, or aggroing a mob by walking too close to it.
Pulling a linked group; some mobs may be linked together, so that when you aggro one, you will aggro a whole group that must be fought simultaneously.
Healing or applying a buff on someone who's on a threat list.
Causing damage and applying debuffs on the target. Some debuffs do not cause any aggro or threat value: Hunter's Mark and Mind Vision.
Becoming in range of a boss's "combat pulse". What a boss's combat pulse does is that when you're in proximity of a in-combat boss, you're automatically put in combat and on the mob's threat list as a nil value.
Combat proximity: Some mobs will put you on their threat list at 0 if you're in range of them, which is much like boss's "combat pulse."
Threat may be calculated to the hundredth place; that is, threat value is a whole number but is multiplied by the threat-inducing ability. (Need confirmation. Thought influenced from the Mirror Image Threat Calculations post.)
To prevent mobs from rapidly changing their targets, a player attacking a mob must exceed 110% of the mob's current target's threat, if the player is in melee range; if the player that's attacking is not in melee range, they require 130% of the mob's current target's threat to pull aggro. However, some boss's abilities will depend on the threat list, and not on aggro. That is, if an ability will not happen on the person with the highest threat, someone may be above the tank's threat but has not pulled aggro (Above 100%, but below 110% or 130%). This effect will result in the tank gaining the ability, and the player highest on threat not gaining it, which may cause a change in strategy or maneuvering. Normally, threat will never decay over time, unless the mob has left combat with you; you may, however, use a threat reducing ability.
If you do exceed 110/130% of the mob's target's threat, you may not actually pull aggro. (Test required. For example, if the aggro-holder has 1000 threat and you have 900, and the aggro-holder reduces threat to below 900 but you do not perform any abilities, you will not gain aggro.) You must do something to actually notify the mob of your surpassing of threat. Also, if a mob's current target dies, it will target the next highest person on threat. This is why if you pull aggro, and you die, the target may not go back to the tank.
When you heal someone, you will cause .5 threat per amount healed, when you have no threat modifiers. Over-healing is not counted, and this threat is only added to the mobs that are tracking the healed target's threat. Healing threat is also split per mob that is tracking the threat; so a healing threat calculation, with no threat reductions, looks like such:
([Effective Healing] * .5) / [Number of Mobs] = Threat per Mob
Buffing a target that's on a threat list and gaining an amount of power (Mana, Rage or Energy) will also cause a small amount of threat. Rage gain is approximately 5 threat per Rage; Energy is unknown; and Mana is .5 threat per Mana gained. There are some exceptions, like Blessing of Wisdom, Adrenaline Rush, and damage-caused Rage, which are excluded because they are passive; but the mana gain that causes threat includes potions. These three types of threat antagonizers are all split between the mobs they affect, and the mobs they affect have no range limit. Overwriting a buff does not cause any more or less threat than if it hadn't overwritten.
Threat modifiers are mechanics that increase or decrease your threat caused, per point of threat. Threat modifiers are multiplicative; so, when you see something like: "Reduces your threat by 30%", it means "Multiplies your threat by .7". Alternatively, you can reverse threat modifiers to see how much damage you gain by using threat modifiers.
These examples show how threat modifiers are calculated, how threat is determined, and how your damage is increased with threat modifications:
1 - [(1 - X) x (1 - Y) x (1 - Z)] = Threat Modifier
X, Y, and Z mean any threat modifications, like Arcane Subtlety, or the .5 power gain modifier for mana. X, Y and Z doesn't mean that you can only have three modifiers; it's just an example.
[Event's Damage, Healing, Gain, etc.] x [Threat Modifier] x  (confirmation needed; see above)= Threat Gained.
If the threat you caused is split mob-wide (I.E. healing threat), you would divide your "threat gained" value by the number of mobs you have.
Rough gain from threat modifiers:
1/[Threat Modifier] - 1 = Damage Gain from Threat Modifiers
Taunt abilities are moves that include: the Warrior ability, Taunt; the Druid ability, Growl; the Paladin abilities, Righteous Defense and Hand of Reckoning; and the Death Knight abilities, Death Grip and Dark Command.
Taunt does three things:
Taunt debuff: The mob is forced to attack you for 3 seconds. Later taunts by other players override this.
You are given threat equal to the mob's previous aggro target, permanently. Importantly, you won't necessarily get as much threat as the highest person on the mob's list, only as much as whomever was being focused on before the taunt debuff (see above).
You gain complete aggro on the mob at the instant you taunt. Usually you would need 10% more threat to gain aggro, but a taunt now gives you instant aggro on the mob. Of course if other people are generating significant threat on the mob, they could exceed your threat by more than 10% before the taunt debuff wears off, and will gain aggro as soon as it does. There is no limit to the amount of threat you can gain from Taunt.
Paladin Taunt: Righteous Defense does the same things, only with three mobs targeting your allied target.
[*]Death Grip: Also drags the target to the casting Death Knight, interrupting spell casting if the target is not already in melee range.[*]Vigilance: If the target on whom this ability is placed is hit with a [u]melee[u] (confirmation needed) ability, Taunt is immediately refreshed for the casting Warrior. No diminishing returns (confirmation needed).
Every class in the game has their own threat modifications, aside from the general ones that apply to everyone. The values listed are for the current [b][u]TBC (need WotLK values)[u][b] max ranks. Spell descriptions only include threat aspects.
Passive: .71 modifier
Vanish: Complete threat reset
Feint: -1050 threat reduction permanently
Tricks of the Trade: For six seconds after using an ability, and including the ability, all threat generated by those abilities is transfered to the target of Tricks of the Trade.
Silent Resolve: .8 modifier
Shadow Affinity: .75 modifier
Fade: Complete threat reduction for the duration of the buff. Threat regained when it dissipates.
Prayer of Mending: The threat is increased according to and is given to the person healed
Arcane Subtlety: .6 modifier (Arcane spells)
Burning Soul: ..9 modifier (Fire spells)
Frost Channeling: .9 modifier (Frost spells)
Invisibility: Complete threat reset, if you completely fade. The tooltip suggests that as you fade, you lose threat for every second you fade, ending at compl-ete threat removal when the fade is finished. Need confirmation for this.
Mirror Image: Complete threat removal for thirty seconds; the Mage Fade. Just as with Fade, you still gain threat that will be significant after Mirror Image dissipates. For more information, view Elitist Jerks: Mirror Image - threat calculations.
Cat Form: .71 modifier
Bear Form: 1.3 modifier
Feral Instinct: 1.15 modifier
Subtlety: .8 modifier
Cower: -1170 threat reduction permanently
Bear: Mangle: 1.3 modifier
Lacerate: 285 added threat
Maul: 322 added threat
Nature's Fury: .7 modifier
Feral modifiers are additive currently, and thus as a bear you'll only have a 1.45 modifier for threat. This is unlike the current warrior modifiers that are multiplicative (and thus gain a modifier of 1.495).
Improved Drain Soul: .9 Affliction modifier
Destructive Reach: .9 Destruction modifier
Master Demonologist: .8 modifier with Imp
Soulshatter: Reduces all threat of all mobs in 50 yards by 50%
Searing Pain: 2x modifier
Some curses and spells generate threat on cast
Feign Death: Complete threat reset on all mobs, if not resisted
Distracting Shot: 900 threat gain
Misdirection: For thirty seconds, the next three attacks done will transfer all threat to the targeted party or raid member.
Omen Information: Omen is a threat meter that tracks all abilities done by people with the add-on, and compiles a list of what the assumed threat of the players are. This add-on is very important for a tank to have; without the tank having it, it's almost useless.
Thanks to Whiteknight and Kenco; most of the information in this article is taken from Whiteknight's post, Kenco's Guide to Threat, which was written by Kenco.
[b][u]Values and mechanics still mostly outdated for Wrath of the Lich King. Please PM Spazzball, or post below, to contribute![u][b]
Paladin's Righteous Fury needs to be updated. It's now a flat 1.9 Holy modifier. The talent no longer increases threat generation. Blessing of Salvation also no longer exists.
Note that in addition to the 90% threat bonus to holy damage, Righteous Fury also provides a hidden buff that increases all threat you generate by 43%. (This is the same as reducing everyone else's threat by 30%, which is in fact exactly what it's intended to do, replacing the old Blessing of Salvation.) This works multiplicatively with the holy damage threat boost, so in fact Holy damage generates 172% more threat with Righteous Fury than without. Needless to say, this is a huge difference, and if you try to tank without Righteous Fury up, you'll notice the difference pretty quickly.
Quite a lot of the information on this table is outdated. A few examples being that fade wipes all threat for the duration of the spell (restoring it afterward), as well as Disengage being completely redesigned (doesn't have anything to do with threat anymore). I'm relatively certain there's a few other mistakes on the list as well, but those were the only two that stuck out at me off the top of my head.
Updated all that I know of in the main post. Sorry for the delay and thanks for the reminder. I'm unaware of the other class-specific modifiers, so I'll need those to be tested and either sent to me or posted here in order to be updated in the post.