This article contains spoilers. Read at your own risk!
It might spoil your gaming experience when first playing the related games.
The Vault, also referred to as The Box, is a massive floating structure created by the Combine before the completion of the Citadel. Located deep within the Quarantine Zone, an area of City 17 that is under lockdown and separated from the rest of the city, it is an important and highly kept secret structure to the Combine, and the purpose of the Vault is one of the major plot points of Half-Life: Alyx.
The Vault is an enormous metallic structure that is suspended high up in the air, with many long and heavy wires protruding from the sides of the structure. The Vault's design is of combine architecture, purposely built to be large and imposing, with 3 points of the structure extending from the middle. The floating structure directly oversees a Combine base which is located beneath it, and as well a large tower that is used as a docking station, with several features, for said structure.
The Vault serves as a technologically advanced prison, of immense importance as The Vault is actually housing and imprisoning one sole being of dangerous status; the mysterious, but highly dangerous and powerful entity known as The G-Man. At an unknown point in time, the Combine was able to detect and discovered the whereabouts of G-Man, who was found inside of an apartment building. The Combine, viewing the G-Man as a immense threat of massive proportions, constructed the Vault that surrounded the entirety of the Apartment building in order to capture and contain the enigmatic entity.
Due to the nature of the prisoner that's being held within, the Vault is of immense important value to the Combine, perhaps more so than Nova Prospekt. The very existence of the Vault is kept hidden and its is very important, that any unauthorized individuals even witnessing the picture of the Vault, is immediately executed. The Vault's location, being the Quarantine Zone, is an area separated from the rest of the City that is infested with alien wildlife, ranging from Zombies to Antlions, which deters any threats. The Vault is heavily guarded, as the floating structure garrisons a large amount of Combine Soldiers, with defending Combine force stationed on the ground immediately beneath the Vault.
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.
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 aloft by several substations that connect to it with 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.