Every time players swing at a monster, they have a chance to critical hit, a rate. If they successfully critical, the monster will take additional damage. This boost to damage is represented in two phases, an increase in pDIF, or effective Attack, and direct modification by Critical Attack Bonus and other similar abilities. Additionally, Critical Hits forced by Sneak Attack and Trick Attack with Thief main receive base damage bonuses from Dexterity and Agility respectively.
For many years, Critical Hits were an appreciated mechanism that held little strategic importance due to the lack of Critical Hit Rate and responsible Dexterity options. This all changed with Scars of Abyssea, Critical Attack Bonus, and Dead Aim, where Critical Hits became a major focus for damage-dealing jobs. This also became something of a bone of contention for some damage-dealing jobs, who lacked the appropriate Weapon Skill options to take advantage of this shift.
As alluded to in the previous paragraph, though all normal swings have the potential to critical, Weapon Skills can only critical hit if their descriptions read "chance to critical hit varies with TP." This has been proven multiple times, though the easiest verification method is using 1-hit weapon skills and looking for damage that indicates a critical hit has occurred. It also appears that some weapon skills, though they possess the appropriate description, do not receive the benefit of base and gear Critical Hit Rate, like Blade: Rin.