Short answers, pulled from the story.
James Clerk Maxwell published an early form of the equations in 1861 and 1862. The modern compact formulation used today is credited to Oliver Heaviside, who reformulated Maxwell's original twenty equations into the rotational vector calculus form that became standard.
Maxwell's equations unified the previously separate theories of electricity, magnetism, light, and associated radiation into a single mathematical framework.
By combining the known values of the permittivity of free space and the permeability of free space, Maxwell derived a wave speed that matched the speed of light. This led him to propose in 1861 that light is an electromagnetic wave.
The microscopic version relates fields to all charges and currents, including atomic-scale contributions. The macroscopic version introduces auxiliary fields that average over atomic detail, making it practical for calculations inside materials, but it requires experimentally determined constitutive relations to close the system.
Since the mid-twentieth century, Maxwell's equations have been understood as the classical limit of quantum electrodynamics. They cannot describe phenomena such as photon-photon scattering, the photoelectric effect, Planck's law, or quantum entanglement of electromagnetic fields.
If magnetic monopoles were found, both Gauss's law for magnetism and Faraday's law would need modification. The resulting four equations would become fully symmetric under the interchange of electric and magnetic fields, a symmetry the current equations do not possess.