Carbuncle
Carbuncle is a bunny/fox-esque companion summoned by Arcanist and Summoner.
General information
Arcanist and Summoner can spawn their pets with the actions 
Summon Carbuncle, 
Summon Ruby, 
Summon Topaz, and 
Summon Emerald. Each type has a different spell style and associated element. The three Summoning Mastery upgrades the Summoner receives between level 30 and level 45 change the Summoner's pet from a Carbuncle into various Egi. Similarly, if an Arcanist becomes a Scholar, they no longer summon a Carbuncle and instead summon a healing faerie.
Alphinaud has a special Moonstone Carbuncle, that heals with the same spells as Eos.
Lore
“Using geometric symbols to lend shape to living aether, certain mages can unlock the latent power within certain gemstones-a power which can be directed to the "summoning" of aetherial beings known as Carbuncles. The inherent traits displayed by these familiars differ depending on the stone used in their creation- emerald and topaz being the most common, with ruby and diamond also proving popular.
— Encyclopædia Eorzea vol 1, page 301
Carbuncle is a term for a red gemstone, such as Garnet. In this case, it refers to the gem in the head of the creature.
Egi Glamours
Summoners and Scholars can give Egi a whole new look with the Egi Glamour text command which gives their pets the appearance of Ruby Carbuncle, Emerald Carbuncle, and Topaz Carbuncle. Egi Glamours are unlocked by completing the Feature Quest 
An Egi by Any Other Name (for Summoner) or A Faerie Tale Come True (for Scholar).
Additional Information
Carbuncle is also a server in the Japanese Elemental Data Center.