Maxwell created a set of four equations, all written in the mathematics of calculus that describe everything there is to know about electromagnetism. Einstein revered it as one of the greatest intellectual achievements of all time.
These equations can be written in either differential form or in integral form.
The first two equations already existed before Maxwell's efforts, and are known as Gauss's law for electric fields and Gauss's law for magnetism. The former states that the total electric flux passing through a closed surface is proportional to the quantity of electric charge inside the surface. The latter states that the total magnetic flux through a closed surface must be zero (and forever rules out the possibility of magnetic monopoles).
Equation three is Faraday's law of induction in concise mathematical form (something that was impossible for Faraday to accomplish). It tells us that a changing magnetic field will produce an induced voltage in a conductor. Specifically, the magnitude of the induced emf in a closed circuit is proportional to the rate of change of the magnetic flux enclosed by that circuit.
The fourth equation begins with Ampere's law, and is the most complex. In it, he tried to create counterpart to the third equation, to should that the magnetic field induced around a wire was proportional to the current in the wire. However, when he could not get this formula to produce universally correct results for all types of fields, he "invented" a second term in the equation (called displacement current). It effectively showed that a magnetic field could be produced by a changing electric field.
This was the first time in the history of physics that an abstract quantity was created for the sake of symmetry. Maxwell was predicting that a changing electric field produced a magnetic field and that a changing magnetic field would produce an electric field. This allowed for self-sustaining electromagnetic waves to exist is empty space, and opened the door to the development of radio!
What Maxwell did, was to take a field of study that was little more than a collection of equations whose relation to one another was unclear, and weld them into one of the truly self-consistent theories of all time.