Familiars are special pets that are available exclusively to Beastmasters. They are called to a Beastmaster's side using special items, commonly referred to as "jugs," and Call Beast, a Beastmaster job ability.
Familiars offer Beastmasters reliability and convenience in exchange for power and/or endurance. While a Familiar is almost never as strong as a charmed monster of the same level, Familiars can be called instantly and stay for a set amount time (typically a very long time) before disappearing - they will never fail to be called or "decharm" and attack the Beastmaster like charmed monsters.
Familiars also allow the use of Snarl, a job ability, and Ready, a pet command, which give the Beastmaster finer control over the flow of battle and increase the survivability of both the Beastmaster and the pet.
However, since Beastmasters tend to rely on the disposability of pets, most Beastmasters only use Familiars if local monsters are not available, as the relatively long recast of Call Beast and the cost of using jugs make Familiars less disposable than charmed pets.
In Final Fantasy V, captured monsters would be placed in "Jugs" and when released would make a specific attack before vanishing, similar to a limited form of summoning and Blue Magic.