Over the course of the eighteenth and nineteenth century, physicists developed several laws describing the behavior of the electric and magnetic forces.
The first is the relationship between electric charge, force, and distance, discovered by Charles-Augustin de Coulomb (1784). This is typically re-written in a form derived by Carl Friedrich Gauss (1867), and referred to as Gauss's law.
The second is the relationship between a current and the magnetic force. Part of the relationship was first discovered by Andr'e-Marie Amp`ere (1826); Amp`re's law applies to very specific configuration of currents, such as infinite straight lines (e.g., long wires) or planes.
The third is a mathematical representation of the fact that there are no magnetic monopoles. It is also named for Gauss, because it's mathematical form is related to the first law. The second and third laws are generalized as the This law is generalized as the Biot-Savart law for steady currents, in honor of the realization of Jean-Baptiste Biot and Felix Savart that a current would cause a magnetic needle to be deflected (1820).
The fourth is the law of induction, in which an electric field is generated by a magnetic flux that changes with time. It was discovered independently by Michael Faraday and Joseph Henry in 1831, and is named for Faraday, who published the result first.
James Clerk Maxwell (1861) realized that Ampere's law was not self-consistent, and added a term relating the magnetic field and currents to an electric field that changes with time. With Maxwell's adjustment, the four equations now revealed that a changing electric field could produce a magnetic field, just like how a changing magnetic field would produce an electric field. Moreover, the equations can be rearranged to reveal the existence of a wave that could propagate through space at the speed of light.
When Maxwell wrote all four equations down together, he realized that his theory predicted the existence of a wave that traveled at the speed of light. Thus, light was hypothesized to be a traveling wave composed of oscillating electric and magnetic fields. This was confirmed dramatically by Heinrich Rudolf Hertz (1888), who demonstrated that the waves existed by building an experiment that produced and detected radio waves.
Faraday's law of induction led to the development of transformers, inductors, motors, and generators.
Electromagnetism also predicted that light could be produced by accelerating charges, which led to the discovery of radio waves. Therefore, the theory is directly responsible for (over-the-air) radio and television transmission, cellular phone systems, satellite communication, radar, microwave ovens, fiber optics, and a host of industrial processes.
Einstein was inspired to develop the theory of special relativity by electromagnetism. He noticed that two laws of electromagnetism produced the same practical results: Faraday's law describing how a changing magnetic field produces an electric field, and Hendrik Lorentz's law (1892) for how a moving charge is deflected by a magnetic field. He explained this relationship by postulating that light moved at a constant velocity, regardless of how fast an observer was moving. Thereby, he unified space and time into one set of equations.
The idea that moving electric charges radiate light introduced a paradox into atomic theory at the turn of the 20th century. Resolving this quandary helped spark the development of quantum theory.
The standard introductory textbook on this subject is Introduction to Electrodynamics, by D. J. Griffiths (1989, Prentice Hall, Inc., Englewood Cliffs, NJ).
See the Wikipedia articles on Coulomb, Coulomb's law, Gauss's law, the Biot-Savart law Felix Savart, Faraday's law of induction, Heinrich Hertz, and