FF1 Places

Dragon's Cave, Cardia Islands (1n): The home of the Dragons and Bahamut.
Cornelia, Coneria (1n): The large town where you start also has a large castle whose princess has been captured by an evil knight.
Crescent Lake: The home of Lukin and the Circle of Sages.
Earthgift Shrine (GBA/PSP Only): Unlocked after defeating Lich in the Earth Cave, this dungeon of the Soul of Chaos only has 5 floors with a possibility of facing bosses from FF3's Dark World.
Elfheim, Elfland (1n): Home of the elves. Duh.
Gaia: A town in the north houses a fairy's spring.
Hellfire Chasm (GBA/PSP only): Marilith's Soul of Chaos, the Hellfire Chasm has 10 randomized floors and has possible fights with the four fiends of FF4 (Rubicante, Scarmiglione, Cagnazzo, and Barbariccia) as well as the fourth ribbon of the game.
Ice Cavern, Ice Cave (1n): The sleeping place of the Floater stone.
Labyrinth of Time (PSP only): First available after defeating the Four Elemental fiends, the Labyrinth of Time houses multiple levels of frustrating puzzles, powerful items, and the boss Chronodia.
Lifespring Grotto (GBA/PSP only): This Soul of Chaos is unlocked after defeating Kraken, and contains arguably the hardest bosses (including Ultima Weapon) of the four dungeons but yields the best shield in the game: the Hero's Shield.
Lufenia, Lefein (1n): This town is populated with ancient warriors who don't speak English.
Marsh Cave: The hiding place of the Crown.
Matoya's Cave: Self-explanatory, I think.
Melmond: A town whose earth and people are rotting away. Dr. Unne lives here.
Tower of Mirage, Mirage Tower (1n): Getting to the top of this lets you reach the Flying Fortress.
Mount Duergar, Dwarf Cave (1n): Home of the dwarfs, who build canals and make swords.
Mount Gulg, Gurgu Volcano (1n): Home of Kary, the Fire Fiend.
Onlak, Onrac (1n): This town has a submarine leading to the Sunken Shrine.
Citadel of Trials, Ordeal Castle (1n): Complete this to get the Rat's Tail. Compare with Mount Ordeals (4).
Pravoca, Pravoka (1n): A port town which has been overrun by pirates.
Sage's Cave, Sarda's Cave (1n): The home of Sarda the Sage, who'll give you a Rod.
Sunken Shrine, Sea Shrine (1n): This is the home of the mermaids and of Kraken, the Water Fiend.
Flying Fortress, Sky Castle (1n): The home of Tiamat, the Air Fiend.
Temple Of Chaos, Temple of Fiends (1n): The home of Garland the errant knight who kidnapped Sarah is also the gateway to 2000 years ago by way of a black Orb. At the end of it resides Garland/Chaos.
Terra Cavern, Earth Cave (1n): Lair of the Vampire and Lich, the Earth Fiend.
Titan's Cave, Titan's Tunnel (1n): Titan guards this passage to Sarda the Sage.
Waterfall Cavern, Waterfall (1n): A robot here will give you a Cube.
Western Keep, Northwest Castle (1n): The home of Astos.
Whisperwind Cove (GBA/PSP only): Unlocked after defeating Timamat, it is the deepest of the Souls of Chaos, weighing in at over 40 floors, but is the most linear of the group.