This experiment was carried out in 1887 by Albert Michelson and Edward Morley. Michelson is known, among other things, for a remarkably accurate measurement of the speed of light in 1879, winning the Nobel Prize in 1907, and inventing the interferometer. His experiment with Morley was designed to prove the existence of the Ether. Since waves in water need a medium to move in (water) and sound waves do as well (air), it is logical that light also needs a medium to move in. This is particularly so because waves behave in exactly the same fashion. This is true whether they are sound waves, light waves, water waves, radio waves, or any other kind of wave. And light waves move even in a vacuum, so the case for an otherwise undetectable medium that light waves travel through was, and is, very strong.

Simple.wikipedia.org describes the experiment in this way: (They use “aether” rather than Ether.)

The Earth travels very quickly (more than 100,000 km per hour) around the Sun. If aether exists, the Earth moving through it would cause a “wind” in the same way that there seems to be a wind outside a moving car. To a person in the car, the air outside the car would seem like a moving substance. In the same way, aether should seem like a moving substance to things on Earth.

Their experiment was designed to measure the speed and direction of the “aether wind” by measuring the difference between the speed of light travelling in different directions. It measured this difference by shining a beam of light into a mirror that was only partially coated in silver. Part of the beam would be reflected one way, and the rest would go the other. Those two parts would then be reflected back to where they were split apart, and recombined. By looking at interference patterns in the recombined beam of light, any changes in speed because of the aether wind could be seen.

The experiment produced a “null result”. This was a disappointment to Michelson, since he was sure of the existence of the Ether, and even after this experiment he continued to research it. Many other scientists, notably Einstein, took the result as proof that the Ether did not exist. Michelson remained interested in the other possibilities suggested by his result. One was that the Ether near the Earth was dragged along by it. The other was that the Earth was not orbiting the Sun.