Short answers, pulled from the story.
Analytic geometry was independently invented by Rene Descartes and Pierre de Fermat in the 17th century, though Descartes is sometimes given sole credit. The field is also called Cartesian geometry after Descartes. The 11th-century Persian mathematician Omar Khayyam and the ancient Greek Apollonius of Perga are considered important precursors.
Descartes introduced analytic geometry in an essay called La Geometrie, published in 1637 as an appendix to his Discourse on Method. The work was initially poorly received due to gaps in its arguments, but gained recognition after van Schooten translated it into Latin and added commentary in 1649.
Fermat always began with an algebraic equation and then described the geometric curve satisfying it, while Descartes started with geometric curves and derived their equations afterward. Fermat's manuscript circulated in Paris in 1637, just before Descartes published his Discourse on Method.
Omar Khayyam, an 11th-century Persian mathematician, identified the foundations of algebraic geometry and wrote the Treatise on Demonstrations of Problems of Algebra in 1070, which laid principles of analytic geometry. His geometric solutions to general cubic equations helped close the gap between numerical and geometric algebra.
The main systems are Cartesian coordinates, which use ordered pairs or triples of numbers; polar coordinates, which use a distance and an angle; cylindrical coordinates, which add a height to polar coordinates; and spherical coordinates, which describe points by a distance from the origin and two angles.
Analytic geometry studies geometric shapes using a coordinate system, representing them with numerical equations and extracting numerical information from those representations. Synthetic geometry, by contrast, studies geometric relationships without using a coordinate system or algebraic equations.