Answer: The Michelson-Morley experiment showed that the speed of light is constant. Using the Earth as a big flying laboratory, Michelson and Morley measured the speed of light as it was projected in the direction of the earth's movement through the ether and light's speed as it was projected perpendicular to the direction of movement. If ether existed and affected light, the speed of light should have been different. As it turned out, they found the speed of light was constant in whatever direction they shined their flashlights, regardless of the earth's path through the etherư.

The speed of light was measured using a fairly simple apparatus that displayed interference patterns.

The results of Michelson-Morley were not satisfactory explained until Einstein enlightened the world 18 years later.

For a nice description of the physics, see the first volume of the Feynman Lectures on Physics [1].

BONUS: The Millikan Oil-Drop experiment measured the charge on an electron. This was accomplished by balancing a tiny charged oil drop in an electric field. Gravity pulled the oil drop down, the electric field opposed gravity. The oil drop was charged with X-Ray bombardment. The experiment was published in 1924 [2].

[1] Feynman, Leighton, Sands, The Feynman Lectures on Physics, California Institute of Technology, 1989, pg 15-3.

[2] Robert Millikan, The Electron, University of Chicago Press, 1924, 2nd ed.