WoW gear score
Gear score is a numerical value based on the statistics of a player's gear. It is (supposedly) the maximum potential of a player's performance. However, it is up to the player to perform that well. In addition to item level (ilvl), the item's actual statistics or abilities may or may not be included for calculation, as determined by specific formulas or addons.
Most commonly, gear scores are a metric used to evaluate the relative strength of a player's equipment as a way of determining if that player meets a given threshold to join in certain raids or groups going through an instance. It should be noted that a given gear score, no matter which formula is used, does not necessarily mean that other players can not demand higher scores from others before joining a raid or group.
Despite the possible flaws in the system, gear score can be a useful tool in determining an individual player's readiness for certain content. Rather than risking the waste of time and money on wipes and repair bills, gear score can provide a basic indication that a player is likely to possess gear sufficient to run (or even trivialize) the content in question. There is no substitute for checking a player's choice of gems, enchants, glyphs, and spec, plus knowledge of appropriate caps (hit, haste, and/or crit), all of which reveal far more about a player than for what gear he or she has managed to win rolls or purchased by running heroics.
That said, when used as an initial assessment of a player, followed by inspections of their Armory profile (which does display things like item enhancements and achievements), gear score can allow the quick and efficient assembling of groups and raids intending to take on difficult content. Be warned: reliance on gear score, without following up the initial assessment with the armory or inspecting, is in fact counterproductive. It degrades the community by encouraging less knowledgeable players to "upgrade" based strictly on item levels, as well as decreasing the motivation of some players to improve (as they receive raid invites and rolls on loot without needing to do so).
Controversy
The usage of gear scores has caused a large amount of controversy due to a proportion of players who view it as unnecessary and distracting in the face of others options like DPS or Achievements.