Final Fantasy III is a large game. While the world itself may not be as fleshed out as its predecessor, III was the first game to employ multiple world maps, which IV, V and VI would follow in doing (albeit one-way for the latter two).
As for III itself, it can be split 2, technically 3 world maps: The Floating Continent and the Surface World (which has two different states). Unlike Final Fantasy II, FF3 doesn't have much in terms of clearly defining states or territories. Thus, the "sub-sections" of each area are designations that I made up for easier groupings (unless stated otherwise).
The Floating Continent is where the story of Final Fantasy III begins. What looks to be a standard world as in any other RPG turns out to take place on a single part of it. The Warriors of Light spend the first half of the game exploring it and solving various problems that they encounter, albeit without much of an idea of the overall forces at play.
Curiously, the Continent also has a bunch of non-human creatures including Gnomes, Dwarves and Faries. There really isn't much explaining the presence of these creatures outside of implications of each version's respective canon.
The Floating Continent's true nature as revealed in the late game differs depending on the version you're playing.
Sasune: The very starting point of the adventure. Is composed of Inner and Outer regions, seperated by the Parmeni Mountains. Consists of Ur, Kazus, Castle Sasune, Canaan and Mt. Jenora.
Beast villages: Grouping together all of the villages which have non-human creatures living in them; Namely Tozus, the Living Woods, Gulgan Gulch and the Dwarven Hallows.
Argus & misc: Consists of the territories associated with or in the vicinity of Argus: Castle Argus, Tokkul, the Vikings' Cove, the Tower of Owen and the Village of the Ancients. For simplicity's sake, Gyshal is included here.
After defeating Hein, the party turns their boat into an airship, allowing them to leave the Floating Continent and reach the surface. Prior to reawakening the Water Crystal, the surface world is but a massive sea, with only a few locations still remaining. After the Water Crystal regains its light, however, the rest of the surface is exhumed and life carries on as normal.
The Boundless Ocean: The state of the world after the flood of darkness is unleashed. Consists of the Wrecked Ship, the Water Temple and the Cave of Tides.
Amur: The town that the Warriors of Light awaken in after restoring the world to normal. Has a sewer and is in close proximity to Goldor's Manor.
Job Villages: a catch-all for all of the job-related towns in the game, namely Duster, Replito and Falgabard.
Saronia: The largest empire in the world, boasting a castle and four different districts.
Dalg Continent: A continent with high winds that serves as the hideaway for Doga and his apprentices.
Syrcus: The Crystal Tower (i.e. Syrcus Tower) and its surroundings/associated areas.