Vault Dweller

In a general sense, a vault dweller is just someone who lives in one of the Vault-Tec Vaults.

When capitalized, the Vault Dweller is the main character of Fallout. He was sent out by the Vault 13 Overseer on a misson to find a replacement for the Vault's broken water chip. In the course of his adventure, he ended up defeating the mutant threat (by blowing up an abandoned military base overrun by the super mutants and housing massive FEV stockpiles) and assassinating the Master, and the leader of his cult, the Children of the Cathedral's Morpheus.

At the end of Fallout, having completed his task and returned home, he was exiled from Vault 13 by the Overseer (as it was thought that his contact with the outside world would cause disruption within the vault). He wondered northwest of Vault 13 and founded the village of Arroyo where the Chosen One, his grandchild and main character of Fallout 2 was born.

Note that while in Fallout the player could choose to be either male or female, the canon Vault Dweller, as established by Fallout 2, was male. Also, the ending cinematic in the original Fallout always portrayed a male (or at least male-looking) character walking away into the desert, even if your character was female.

Appearances in games
The Vault Dweller is the player character in the first Fallout. He does not appear in Fallout 2, but is mentioned in every Fallout game. He also makes a cameo in Fallout: Brotherhood of Steel and is an unlockable player character.