Short answers, pulled from the story.
A pointed arch, also called an ogival arch or Gothic arch, is an arch whose two curving sides meet at a sharp angle at the crown. Its form is derived from the intersection of two circles. It was especially important in Gothic architecture, where it enabled the rib vault, taller walls, and larger windows.
Crude pointed arches from the Bronze Age site of Nippur date to earlier than 2700 BC. The Byzantine Karamagara Bridge, with its pointed arch spanning 17 meters, is a significant pre-Islamic example. The clearest surviving pre-Islamic pointed arches are the two arches of the Chytroi-Constantia Aqueduct in Cyprus, dating to the 7th century CE.
The pointed arch reduced the outward thrust on its supports compared to a semicircular arch carrying the same load and span. Medieval builders used it to create rib vaults, where thin stone ribs channeled weight downward to pillars rather than pressing outward on walls. Flying buttresses added outside the building handled the remaining outward forces, allowing walls to be thinned, raised, and filled with stained glass windows.
The sexpartite vault divided the nave ceiling into six compartments using a transversal pointed arch and required alternating columns and pillars; it was used at Sens Cathedral and Notre-Dame de Paris. The quadripartite vault reduced the compartments to four, distributed weight equally to four pillars, and allowed the vault to span a wider space. Amiens Cathedral, Chartres Cathedral, and Reims Cathedral all used the quadripartite design.
The Tudor arch is a variation of the Islamic four-centred arch, a low, wide pointed form whose sides rise steeply from the springing points before flattening into a broad curve. It was widely used in English architecture during the Tudor dynasty, from 1485 to 1603. It is flatter than the related Persian arch, which is described as moderately depressed.
Gothic Revival architecture brought the pointed arch back in the 18th and 19th centuries after the Renaissance had largely replaced it with classical forms. Horace Walpole began using it at Strawberry Hill House in Twickenham, London from 1749 onward. Later examples include the British Houses of Parliament in London, rebuilt in 1840-1876, and a Gothic train station at Peterhof, Russia, completed in 1857.