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One of my favorite physicists to learn about was James Clerk Maxwell, for the fact that he unified the study of E&M in physics and he would often disprove theories that did not work as a Mathematician.

I remember my physics professor mentioning a disproof Maxwell had done with what was then the current knowledge of how currents flowed or were held through capacitors... This is probably way off from what he actually did disprove since it has been over 5 years since I had been to this particular lecture but could anyone tell me or demonstrate the exact disproof by counterexample that Maxwell had done with capacitance?

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Are you talking about the famous derivation of the displacement current, where Ampère's law is both true and false depending on what surface you choose to integrate through, despite the same boundary, as below:

Current through surfaces S_1 and S_2 are different motivating the introduction of the displacement current

(Image from WikiMedia commons http://commons.wikimedia.org/wiki/File:Displacement_current_in_capacitor.svg)

The solution of this was to add a term to Ampère's equation that depends on the time derivative of the electric field. And then, there is no paradox anymore.

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