Short answers, pulled from the story.
A vault is a self-supporting arched form, commonly of stone or brick, used to cover a space with a ceiling or roof. It becomes self-supporting only once the topmost stone, the keystone, is placed, locking the surrounding voussoirs together so the temporary wooden centering can be removed.
Among the earliest known examples of vaulting is the Neolithic village of Khirokitia on Cyprus, dating to around 6000 BCE. Its circular buildings were roofed with beehive-shaped corbel domes of unfired mud-bricks and also represent the first evidence of settlements with an upper floor.
The earliest known barrel vaults were built by the Sumerians, possibly beneath the ziggurat at Nippur in Babylonia, using fired bricks cemented with clay mortar. In ancient Egypt, the granaries built by the 19th dynasty pharaoh Ramesses II at Thebes are among the earliest examples, with a span of 12 feet.
The pointed arch rib vault first appeared at Cefalù Cathedral, pre-dating the choir aisles of the abbey of Saint-Denis near Paris, which were built by the abbot Suger in 1135. Cefalù is a Romanesque cathedral, meaning its masons experimented with Gothic rib arches before the technique was widely adopted.
Justinian's architects used pendentives, curved triangular surfaces that transition from a square base to a circular one, to support a hemispherical dome over a square floor plan. The dome was raised higher after earlier versions fell, and 40 windows were pierced around its base, creating the impression it was suspended in air. The dome reached an internal diameter of 107 feet.
A fan vault is a type of vaulting found in English late Gothic architecture, built as a single surface of dressed stones with ribs radiating like a fan from each springing point. The earliest example is the east walk of the cloister at Gloucester Cathedral. Fan vaulting is peculiar to England, with no true equivalent on the European continent.