Mog is a moogle that appears in Final Fantasy XIII-2 and Lightning Returns: Final Fantasy XIII. Not to be confused with the Mog of Final Fantasy VI, he originated in the Ocean of Time before a paradox pulled him into the void, in which he wandered until ending up in Valhalla. There, as the weakest creature in the realm, he became a friend and ally of Lightning.
When Lightning encounters Noel Kreiss, she sends Mog along with him to find Serah. Mog accompanies Serah and Noel for the duration of their journey and has the power to turn into a weapon for Serah.
Lightning Returns: Final Fantasy XIII
In Lightning Returns, Mog has become the chief of a moogle village in the forest in the Wildlands. Lightning encounters him here after she rescues the village from a swarm of dryads. Mog tries to flee, not willing to face her after he couldn't do anything to save Serah, but Lightning catches up and asks him to be there for Serah when she returns, as Lightning doesn't know if she herself will be able to be there for her.
On the very last day of the game, Lightning encounters Mog in the Ark instead of Hope. Mog offers the same service in allowing Lightning to purchase items in exchange for EP before she returns to the surface. Mog later reappears in the final dungeon and meets Lightning just outside Bhunivelze's chamber, where he offers what last support he can before Lightning must engage in the final battle.
After Bhunivelze's defeat, Mog briefly reunites with Serah, but then chooses to depart the world along with the Eidolons. However, he promises Serah that they'll see each other again.