Abilities can be divided into 3 categories.
・Command (Main Command/Sub Command)・Support・Reaction
What abilities an unit can acquire depends on his/her jobs.
To unclock&level abilities, you have to spend JP and Gil.
To use abilities in battles, you have to set Sub Command Job, Reaction Ability, and Support Ability.
Abilities have their own dmg/heal multiplier. And it can't be seen in the game. Ability descriptions don't explain how it works.
/u/Krazplay on reddit allowed us to use data collected (really appreciate it!), so that we will add more info to each ability page from now on.
Also some jobs use not only ATK/MAG but also DEX/AGI/Luck/etc... as bonus for their ability damage/heal.
You can check /u/Krazplay's datasheet and /u/deal647's job bonus sheet for more details. Links are on the Damage Mechanics guide.
See below for required JP/Gil to level abilities. Check also JP Guide for how to farm JP.
You can enhance abilities by leveling them. Cast speed gets faster, damage dealt gets larger, the effect of lowering stats is weakened, and the effect of raising stats is strengthened.
Each unit has main commands decided by Job 1. Main commands have an icon that's slightly different from that of sub commands.
UR Units & Tier List
Espers List & Tier List (Summons)
Reroll Guide: Tier List & Tips