There have been two novellas released for Final Fantasy XIII: Episode Zero -Promise- and Episode i. The former serves as a prequel to the game, while the latter is a epilogue that leads into the events of FFXIII-2. There's also a series of sidestories in the Ultimania.

Episode Zero -Promise-

Episode Zero covers the thirteen days prior to the start of the game. There are some events in the novella that don't occur in the game, and vice versa. Personally, I think the novella is a great expansion to the game; it gives us a lot more detail about the characters' lives before everything happened to them, and provides a lot of context to their actions. In the case of Lightning and Snow, especially, it fleshes out their backstories in Bodhum, which in the game is simply understood.

There's also a lot of detail about what Fang and Vanille do when they wake up from their crystal sleep, and how completely alien the world of Cocoon is to them. This isn't depicted as much in the game, in part because they had two weeks to acclimate, but at the start they had no idea how to deal with Cocoon's many automated systems and functions. It's very interesting to read about.

You can read a fan translation of the novella yourself here.

Audio Dramas

Square Enix also put out two audio dramas, titled Final Fantasy XIII Episode Zero -Promise- Fabula Nova Dramatica Alpha and Omega. These contain similar stories to the novella and the stories of the thirteen days, with some new content (like Serah going into the Pulse Vestige).

As I don't know Japanese, I can't appreciate the audio dramas myself; the FF Wiki has more information on them. As far as I know, no one has translated them. You can visit the official site here.

Episode i

Episode i serves as an epilogue to the game, and contains the original timeline of the events of that day. Basically, it gives a look at the immediate aftermath of the ending and what the characters' plans are right after the game ends. It's an important prologue to FFXIII-2, and Episode i was in fact republished as the first part of the first FFXIII-2 novella, Fragments Before.

The novella does have an official translation; it was released as a pre-order bonus for FFXIII-2. You can read this official translation here. If you'd prefer to read the fan translation, you can do so here (text only) or here (with images from the ending FMV).

A fan audiobook of Episode i can be found here.

Final Fantasy XIII Side Story: A Dreaming Cocoon Falls into the Dawn

This is an obscure novella published in the Final Fantasy XIII Ultimania Omega, the official Japanese guidebook for the game. The novella follows a number of characters, some completely new, some briefly mentioned in the game, such as the members of NORA, and details what happens to them following the Purge and the adventures of the main party.

This doesn't have an official translation either, but you can read a plot summary at the FF Wiki, and a partial translation here.