Critical Hit
Affects the amount of physical and magical damage dealt, as well as HP restored. The higher the value, the higher the frequency with which your hits will be critical AND the higher the potency of critical hits. By default, the chance of a critical hit is 5%, with a damage increase of 40%. A critical hit is indicated by the flying text damage number appearing in a larger font, and with an exclamation point (!).
When under an effect that increases the chance of a Critical Hit, actions that are guaranteed to deal a Critical Hit will deal increased damage, according to the Critical Hit rate (as of Patch 6.2).
If an attack is both a Critical Hit and a Direct Hit, it is considered a Critical Direct Hit, and the damage boosts of the two effects are multiplied by each other. A critical direct hit is indicated by the flying text damage number having two exclamation points (!!).
Heals can Critical Hit, but they cannot be Direct Hits.
Certain actions and traits have additional effects upon landing a critical hit, such as 
Scholar's 
Adloquium and 
Monk's 
Deep Meditation II.
Because Critical Hit acts to boost both the frequency and power of criticals, it is widely considered the most important secondary attribute for increasing DPS and as such is often prioritized for materia melding.
History
Critical Hit was formerly named Critical Hit Rate before Patch 4.0 and the release of Stormblood.