Short answers, pulled from the story.
Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Along with arithmetic, it is one of the oldest branches of mathematics. A mathematician who works in geometry is called a geometer.
Euclid revolutionized geometry around 300 BC with the Elements, widely considered the most successful and influential textbook of all time. It introduced mathematical rigor through the axiomatic method and is the earliest example of the definition, axiom, theorem, and proof format still used in mathematics today. The Elements was known to all educated people in the West until the middle of the 20th century.
The earliest recorded beginnings of geometry trace to ancient Mesopotamia and Egypt in the 2nd millennium BC. The earliest known texts are the Egyptian Rhind Papyrus and Moscow Papyrus and Babylonian clay tablets such as Plimpton 322, dated around 1900 BC. South of Egypt, the ancient Nubians established a system of geometry including early sun clocks.
Geometry split into subfields including differential geometry, algebraic geometry, computational geometry, algebraic topology, and discrete geometry. Other branches arise by disregarding properties of Euclidean space, such as projective geometry, affine geometry, and finite geometry. Complex geometry, convex geometry, and geometric group theory are also distinct fields.
The geometry underlying general relativity is a famous application of non-Euclidean geometry. Differential geometry matters to mathematical physics because of Albert Einstein's general relativity postulation that the universe is curved. Riemannian geometry and pseudo-Riemannian geometry are used in general relativity.
Non-Euclidean geometries were discovered in the 19th century by Nikolai Ivanovich Lobachevsky, Janos Bolyai, and Carl Friedrich Gauss. They arose from research on the parallel postulate carried on through figures including Ibn al-Haytham, Omar Khayyam, Nasir al-Din al-Tusi, John Wallis, and Giovanni Girolamo Saccheri. Replacing the parallel postulate yields hyperbolic geometry and elliptic geometry.