James Clerk Maxwell was a famous scientist and mathematician during the mid 19th century. HIs most famous works are in the field of electromagnetism. Maxwell gathered laws and equations of electricity and magnetism under one combined theory, uniting and expanding the works of Ampere, Faraday, Gauss and many more. His theory can be summed up into a set of 4 equations known as Maxwell's Equations:

These symbols may look like gibberish to you (if you don't know vector calculus), but it is more important to understand the concepts behind this theory than the math itself.

Maxwell concluded that electricity and magnetism are different aspects of the same phenomena. His equations predict that moving electric charges create magnetic fields, while moving magnets create electric fields (which is how power plants generate electricity). This electromagnetic force, Maxwell theorized, should be propagated by electromagnetic waves.

Electromagnetic waves are waves that travel through space, but instead of physical oscillating particles, it has an oscillating electric and magnetic field, which are always perpendicular to each other and to the direction of motion:

Maxwell was able to calculate the speed of these waves in a vacuum based on the properties of electricity and magnetism.  He found that this speed was the experimentally measured speed of light. His only conclusion was that light must be electromagnetic waves. This fit well with the wave theory of light. But if, as Maxwell's equations predict, all light travels at the same speed in a  vacuum,  this must mean that there is relative motion between the Earth and the aether.