It has vastly different functions in each title, but tends to focus on debuffs and unique black magic abilities.
Arcanist is a pet-using DPS class with a focus on damage over time abilities with a few direct damage spells such as Ruin. Arcanist additionally has a few support abilities such as the ability to Raise and the potential to heal to a certain extent, though this wanes quickly. Arcanist's weapon of choice is the grimoire.
Its defining trait is its ability to summon pets into battle, namely Emerald and Topaz Carbuncles. The Emerald Carbuncle, obtainable at level 4, focuses on long-range attacks, while Topaz Carbuncle, obtainable at level 15, focuses on tanking abilities.
Arcanist can be upgraded into the Summoner (DPS) and Scholar (Healer) jobs at level 30 after a quest. Arcanist is currently the only class that can promote into two distinct jobs and the player is able to level them both concurrently.
Arcane is a synonym for esoteric. It refers to various secret philosophical and religious doctrines and traditions.