This article contains spoilers. Read at your own risk!
It might spoil your gaming experience when first playing the related games.
The Vault is a structure created by the Combine before the completion of the Citadel and is located in the Quarantine Zone within City 17. The purpose of the Vault is one of the major plot points of Half-Life: Alyx.
The Vault was built around a building specifically to contain the G-Man, who was found inside said building. From the outside, the Vault looks like an enormous floating metal ship.
Inside, several broken rooms can be found, some of which are oriented sideways or upside down, or are mirrored across their roofs. Gravity seems to follow the orientation of each room or sometimes be null (this notably doesn't affect Alyx Vance or the Overwatch Soldiers, who are always oriented relative to the outside world). Static, ghost-like apparitions of humans (perhaps former inhabitants of the building) can be seen inside the rooms. At the Vault's core, there is a large icosahedron structure where the G-Man is locked.
The Vault is powered by energy cables from enslaved Vortigaunts from the Northern Star hotel, as Vortigaunt energy is known to limit or nullify the G-Man's abilities, as well as make objects float.
In the beginning of Half-Life: Alyx, Eli Vance and Russell discover that the Vault is used by the Combine to contain some sort of a superweapon and send Alyx to retrieve it. She finds that the Vault is kept afloats using several substations that connect to it by massive cables. She manages to shut one substation down, and from it emerges a Vortigaunt that was used as a power source. The Vortigaunts then offer their help with shutting down the rest.
After investigating further, Eli Vance concludes that the Vault isn't actually holding a superweapon, but instead it is a prison where the Combine keeps are keeping somebody. As Alyx fights her way to the Vault, she overhears a conversation where a scientist working with the Combine mentions that they're holding somebody who miraculously survived the Black Mesa Incident. This leads Eli and Russell to believe that the Vault is a prison for Gordon Freeman himself.
Finally, Alyx manages to shut down the last substation, letting the Vault crash into the Quarantine Zone. She infiltrates the Vault and finds that the Combine used the Vault to imprison the G-Man, whom she unknowingly sets free.