Damage Reduction

That tickles

Exactly What It Says on the Tin.

Put simply, Damage Reduction (or DR, as it's abbreviated) is an indication of how much physical damage a character sucks up before it actually starts to subtract from their Hit Points.

Though used primarily in Role Playing Games, DR can be found across many different media. In games, DR typically is the very first thing calculated, right after base damage is assigned, and before any multiplicative or additional damages are added into the equation.

In most paper-and-pencil RPGs, this usually tops out at numbers less than 10, since even 30 damage in P&P games is enough to kill most low-to-mid-level, non-fighter characters. Electronic and MMORPGs on the other hand, can easily have DR hitting double, or even triple digits.

One very important to note is that, 99% of the time, DR reduces only physical damage - that means that, typically, there are one or more ways to get around a DR-based Damage Sponge: Elemental Damage almost always bypasses DR, and if Magic Damage is different than Physical Damage, suddenly spamming the hell outta Magic Missile seems like a really, really good idea.

The Trope Maker here is a little hard to place, as many/most Tabletop Games miniatures games use some sort of Armor Rating to reduce damage, but the Trope Codifier is without a doubt Dungeons and Dragons, which uses it to a large extent, and is responsible for the tradition of "magic fire beats DR".

Tabletop Games

 * In GURPS, the primary benefit of wearing/installing armor is reducing damage received. DR is also an advantage that can be purchased by characters, races, etc. One to three points of damage reduction seems to be the "realistic" limit for natural DR, possessed by real animals with thick hides/scales or purchasable by players without needing specific GM approval. Previous versions featured the Toughness advantage, a more expensive DR with a two point Cap specifically for human use, with the base advantage restricted to supers or races.
 * Armor piercing attacks generally take the form of a divisor, reducing DR by half or more.
 * The Damage Reduction advantage also has a host of options to modify it's function, in particular conjunction with Damage Typing. The advantage could be used to simulate anything from thick skin to magical resistance against a given element to an ablative force field that needs recharging.
 * Dungeons and Dragons makes extensive use of this trope. Damage reduction is generally not provided by wearing armor, even magical, and is usually an ability granted to monsters. Said monsters oftentimes have a weakness that bypasses their DR, usually written as "DR Value / Weakness", such as the Superman image above. Some materials or enchantments do provide DR on equipment and some class features also grant it.
 * Hardness is a variant used only by inanimate objects, for when players decide to smash down doors, sunder enemy gear, or otherwise wreck their environment. Most objects have little to no armor class and surprisingly little HP compared to a player character. Hardness makes them difficult to actually damage. In theory, this should prevent anyone from just carving a new door through a wall, because standard attacks won't overcome hardness and will never cause hit point damage; in practice, most characters can easily overcome the hardness of stone and players agree to not abuse those rules most of the time.
 * Arkham Horror features a few options for DR. The mobster investigator has it as his special ability. A few items and spells allow for it, including a variation of Hyperactive Metabolism were food provides DR rather then healing.
 * New Horizon has armor... and specific attacks penetrate the armor, as well as attacks that go overboard.
 * Dark Heresy, Rogue Trader, and Deathwatch all have this in the form of both Armor and Toughness. Very few attacks ignore this, though many weapons have Armor Penetration, which ignores a certain amount of Armor. Their parent game, Warhammer40,000, uses a combination of Toughness Values and Armor Saves, though it's a certain chance to ignore damage, rather than reducing it.
 * Any game with damage soak rolls, such as Shadowrun, either version of the Tabletop Game/World of Darkness, and West End Games' D6 system such as the Star Wars RPG. After establishing the damage of the attack, the thing being attacked rolls to reduce the amount of damage actually taken, in terms of their own innate difficulty to damage. Armour may either provide extra dice or reduce the target number of the roll to reduce the damage, depending on system.
 * The Hero System's version of defenses functions a touch differently. A character can have both Defenses and straight Damage Reduction; Defenses outright subtract damage from the roll and can reduce it to zero, while Damage Reduction applies after Defenses and cuts the damage received by a straight percentage. Damage Reduction is one of the "warning sign" powers in the book; characters are expected to have fairly limited Damage Reduction, such as having it only apply to certain types of damage, and across-the-board Damage Reduction is mostly the domain of supervillains expected to take on entire teams of heroes singlehandedly.