Once you've rescued the Reverend and he's back in the Church in Aristo, you can start to change your party's job classes.
Each class comes with its own stat boosts and skill set, and some of the skills are named differently for each party member (but in essence are still the exact same skill). These skills are notated as "Character Specific" in the accompanying pages.
Each class also has a mastery bonus, but unfortunately these bonuses only remain in effect for as long as you keep that class. (In other words, when you change classes you lose the mastery bonus.)
When you assign a new job class, your stats will change automatically, and you'll start with a "class mastery" of 0%. Each battle will earn you Class Points (CP), which will raise the percentage until you reach 100% and the class is mastered.
At the start of the game, only the basic classes are available. In order to progress, a party member must master one or more basic classes in order to unlock the corresponding intermediate class. Then, they need to master one or more intermediate classes to unlock the corresponding advanced class.
There are 20 classes in all - 7 basic, 8 intermediate, and 5 advanced.
Note: in addition to class skills, each party member also learns a small number of unique skills over the course of the game, which are not included in these pages.