Crit is gamer slang for Critical Hit. A Critical is an attack or other action that gains extra damage or healing, usually by landing very high on the combat resolution table or meeting specific criteria.
Each type of damage has a separate chance to land a Critical Hit. These are the player's Spell Damage Crit Chance, Melee Crit Chance, Ranged Crit Chance, etc. Healing spells can also Crit, resulting in a Critical Heal. Also, some spells or abilities will grant a short duration buff that will cause the next spell or combat art, or all spells and combat arts for a short time, to Crit.
A Crit does 30% more damage or healing than the spell or combat art normally does.
In World of Warcraft, every attack has a chance to crit. A character's chance to crit is divided into Crit and Spell Crit. Normal "crit" refers to all physical attacks. Crits are random events. Every attack is determined separately by standard probability.
Typically, a crit will raise the amount of damage dealt by an attack or an ability by 100% for physical, or 50% for spells. Some classes have talents that may influence this bonus. For example, the hunter's Mortal Shots increases this bonus by an additional 30%, meaning instead of dealing an extra 1.0x damage, it would deal an extra 1.3x damage.
Because crits can be a very important component to dealing (or healing!) damage, many players make it a point to raise their crit chance through the use of Crit Rating and Spell Crit Rating. Physical crit can also be raised through Agility and spell crit by Intellect, although at a much slower rate.
Many classes have skills that trigger based on a crit. For example, Ignite deals additional damage over time when a mage crits a fire spell. Illumination returns mana to a paladin after their healing spells crit. Many other such effects exist.
A character's standard crit rate for both spell and melee/ranged can be viewed in the Character Window.