Short answers, pulled from the story.
A Moogle is a sentient fictional species created for the Final Fantasy franchise by Square Enix. Moogles are white-furred creatures typically sporting wings and a pom-pom-tipped antenna, and they first appeared in Final Fantasy III in 1990.
Koichi Ishii created the original Moogle design during his elementary school days, inspired by his love for koalas. He imagined a white koala with bat wings that could inflate its body to float and fly, and later brought the design to Final Fantasy III at his own request.
The name Moogle is a compound of the Japanese words for "bat" and "mole." The name was translated into English by Ted Woolsey, and an early alternative translation, "Mogri," appeared in the original Final Fantasy Tactics before later remakes reverted to "Moogle."
The signature Moogle vocalization is "kupo," which was introduced in Final Fantasy V in 1992. Earlier games had distinguished Moogles with a "meow" sound before the "kupo" became established.
Yoshitaka Amano added the slit-like eyes and pom-pom antenna for Final Fantasy VI in 1994. Later redesigns were created by Ryoma Ito for Final Fantasy Tactics Advance in 2003, Toshiyuki Itahana for Crystal Chronicles and Mobius, Toshitaka Matsuda for XIII-2, and Yusaku Nakaaki for Type-0.
Mog is a recurring named Moogle character who first appeared as a recruitable party member in Final Fantasy VI. The character has since appeared in Final Fantasy IX as a companion to summoner Eiko Carol, in XIII-2 and Lightning Returns as a helper to Serah Farron, and in multiple spin-off titles including Type-0, Crystal Chronicles, Chocobo Racing, and Record Keeper.