Vault 25 is a mentioned Vault Tec Pre-War vault, located somewhere on the west coast in Fallout: New York 'At world's fair', it's location is rather vague as are the details of what exactly happened there.
The Vault, like Vault 88, was not completed, not 100%. The idea was that the Overseer would receive a terraforming kit to expand the vault and add rooms onto it every couple of months. The twist was that the rooms were pre-planned before the war by Vault Tec themselves, who told the Overseer in his briefing that sticking to the plan would ensure their survival as it was mathematically the best course of action.
What it actually was, was the mathmatically WORST course of action and so, the Vault always found itself struggling for resources. This went on for many years until it got so desperate that the people of the Vault revolted and overthrew the Overseer. They have since abandoned their pre-planned pattern and the Vault has prospered ever since.
Kayleigh Carruthers is the only known Vault Dweller from this Vault. According to her, her Father was a major part in the resistance. Like Vault 85, it has opened up to the outside world and some of its residents have opted to travel.
Fallout: New York: At world's fair (Mentioned)