The story of FFXIII is based around its two worlds: the civilized floating world in the sky, Cocoon, and the vast, uncontrolled lowerworld, Gran Pulse. Hundreds of years before the start of the game, they warred against one another in what is known as the War of Transgression. Now, Cocoon's inhabitants fear and hate anything from Pulse.
The fal'Cie of Pulse and Cocoon warred against each other, and branded the humans of their side to serve as l'Cie and do their bidding. Near the end of the war, two particular l'Cie, Fang and Vanille, were chosen to bear the Focus of becoming the beast Ragnarok and destroying Cocoon. Fearing this fate, Vanille refused to do it, and Fang alone became Ragnarok, leaving a deep wound in Cocoon's shell. Before they could finish their task, they were turned to crystal.
And thirteen days before the start of FFXIII, they awoke.
To Pulse, Cocoon is a "floating nest of vipers," but to those who live there, it's simply home. Tens of millions of people live in Cocoon, served by over eight million fal'Cie to provide for their every need. Cocoon is a veritable paradise, meant to keep its population happy and safe until the fal'Cie have need of them.
Inhabitants of Cocoon live on the interior; Cocoon itself is about the size of North America. [source] Everything inside Cocoon is powered by fal'Cie, from one that serves as the sun (Phoenix) to fal'Cie that open automatic doors. Indeed, it's because of fal'Cie that Cocoon can even stay afloat.
Cocoon has many different cities, a few of which are visited in the game, such as the seaside town Bodhum, the fairground of Nautilus, and the commerce center of Palumpolum. Its capital is Eden, a floating city in Cocoon's sky. Cocoon also has less habitable areas: the Hanging Edge, where the game begins, is where Cocoon was scarred in the war many years ago, and has yet to be fully repaired. There are areas for study and research, such as the Gapra Whitewood and the Sunleth Waterscape.
Cocoon is ruled by the Sanctum, a human branch; fal'Cie prefer to stay out of human affairs and leave the governing to them. You know, if you ignore the fact that the Primarch, the leader of the Sanctum, is a fal'Cie is disguise. Whoops. The Sanctum military is divided into two branches: PSICOM (Public Security and Intelligence Command) and the Guardian Corps. The former is responsible for Pulse threats, while the later is focused on normal peacekeeping. The Wide-Area Response Brigade, also known as the Cavalry, is an airship fleet that's part of the Guardian Corps.
The vast lowerworld is called Gran Pulse by its inhabitants, and simply "Pulse" by those who don't know better. Unlike Cocoon, Pulse's fal'Cie care little for their human inhabitants past occasionally turning them into l'Cie to do their bidding. There were once many people, cities, and civilizations on the surface of Pulse, but by the time of FFXIII, they are nowhere to be found.
Pulse itself is said to be comparable in size to Earth. [source] Several locations can be visited by the party when they reach Pulse during the story, including optional locations. The most notable is the Archlyte Steppe, the largest single area to ever appear in a Final Fantasy at the time of FFXIII's release. Most sidequests in the game start or branch off from this area.