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The term "vault dweller" refers to any current or former inhabitant of one of the government-contracted Vault-Tec fallout shelters known as Vaults. However, the character referred to as "the Vault Dweller" is the protagonist of Fallout. The character can be male or female, though canonically it was a man. He is sent out into the Wasteland on a quest to find a new water chip for Vault 13, for without it, the Vault can no longer recycle water and the inhabitants will subsequently die out in about 150 days. Along the way he makes contact with a wide variety of people who assist him on his quest. However, while completing his task, he learns of an impending mutant invasion, and is sent back out into the Wasteland to prevent it. The Vault Dweller must defeat " the Master," leader of the mutant army, as well as destroy the Super Mutants' military base, so as to prevent an attack on any of the towns in the future. Upon returning to his home, Vault 13, he is exiled permanently into the wasteland by the Overseer, the leader of Vault 13, who claims that the Vault Dweller has "changed too much." Depending on traits and actions the Vault Dweller took in the Wastes he may gun down the Overseer then leave, or simply walk away. After being exiled he wanders with people who left the vault with him and eventually founds the village of Arroyo. He settles down with a woman named Pat. They have a child who grows up to be the Village Elder. After a long time in Arroyo the Vault Dweller decides it is his time to leave and dissapears into the Wastes, leaving only his Vault 13 uniform and Pip-Boy. The Village Elder has a child who would grow up to be the Chosen One, the protagonist of Fallout 2, who would go on to save Arroyo and its Vault 13 relatives.
To read his opinions on all the stuff he had to do here are his memoirs: http://fallout.wikia.com/wiki/Vault_Dweller's_memoirs