Short answers, pulled from the story.
Electromagnetism is the interaction that occurs between particles with electric charge by way of electromagnetic fields. It is one of the four fundamental forces of nature and the dominant force in the interactions of atoms and molecules. It can be thought of as a combination of electrostatics and magnetism.
Maxwell's equations are a set of four partial differential equations that provide a complete description of classical electromagnetic fields. Their discovery culminated in the 1860s, and James Clerk Maxwell unified earlier developments in his 1873 work A Treatise on Electricity and Magnetism. They predicted the existence of self-sustaining electromagnetic waves.
In April 1820, Hans Christian Orsted observed that an electrical current in a wire moved a nearby compass needle, proving that an electric current produces a magnetic field. The CGS unit of magnetic induction, the oersted, is named in his honor. His findings influenced Andre-Marie Ampere's mathematical work on magnetic forces between current-carrying conductors.
The electromagnetic force holds atoms together through electrostatic attraction between nuclei and electrons, and it lets atoms combine into molecules including proteins. It is responsible for the forces we feel when pushing or pulling objects, which arise from intermolecular forces. It also underlies electrical energy, communication, sensors, computation, and motors.
Maxwell's equations made the speed of light in vacuum a universal constant set by the permittivity and permeability of free space, which violated Galilean invariance. After contributions from Hendrik Lorentz and Henri Poincare, Albert Einstein resolved the conflict in 1905 by introducing special relativity, a new kinematics compatible with classical electromagnetism.
Investigation into electromagnetic phenomena began about 5,000 years ago, when ancient Chinese, Mayan, and potentially Egyptian civilizations recognized that magnetite had attractive properties. Around 600 B.C.E., Thales of Miletus discovered that rubbed amber could pick up light objects and hypothesized a connection to magnetism. A mathematical basis did not emerge until the late 18th century.