The Michelson-Morley Experiment
Classical physicists assumed that there was ether, the medium in which light is supposed to travel. Two American experimental physicists, an experimental physicists designs apparatuses, conduct experiments and does other things which don't require a lot of thinking (at least less then theoretical physics, which is nothing but thinking, Albert Michelson and Edward Morley, set out to design a device to measure the speed of light relative to the ether.
Using a very accurate measuring technique (nowadays called "Michelson interferometry"), they set out to do the experiment. To describe the experiment would require a bit of imagination on your part; you're just not used to hearing these things in today's world!
The ether, according to Newtonian mechanics, is always at rest. According to modern theory at that time, the speed of light would change in the ether when measured by anyone in motion. Now, the earth also moves in absolute space; it however, changes direction. It moves in one direction in January, the opposite direction six months later, in June. Therefore, we human beings on earth should measure the speed of light to be different in different directions.
Michelson used this idea to test the theoretical physicists (ones with brains) predictions. According to the experiments he conducted he could find no variation in the speed of light in different directions whatsoever. The speed of light turned out to be the same, in all directions and at all times, both in 1881 in Postdam, Germany and later in Cleveland in an experiment that Michelson performed with chemist Edward Morley.
Theoretical physicists were very skeptical of the results of the experiment. Interesting experiments are often very difficult to perform and it's easy to make one small mistake that goes unnoticed by everybody, yet disrupts the results of the whole experiment. Yet, some experiments, when conducted, challenge our basic understanding of the Universe and how it behaves. The Michelson-Morley experiment was one.
Several theoretical physicists tried to find an error in the experiment. They looked closely at the experimental apparatus and the technique in which the experiment was done. After all this examination, the experiment began to emerge from the "black listed experiments" to the list of "proceed with caution".
An Irish physicist, George F. Fitzgerald was the first to accept the Michelson-Morley experiment, and he speculated its implications. By comparing it with other experiments he found that there is something wrong with physicists' conception of length, and therefore, that there is something wrong with Classical Newtonian and Galileian mechanics and relativity. In his short 1889 article, in the American journal Science, he wrote:
"I have read with much interest Messrs. Michelson and Morley's wonderfully delicate experiment . . . Their results seems to be opposed to other experiments . . . I would suggest that almost the only hypothesis that can reconcile this opposition is that the length of material bodies changes, according as they are moving through the ether [through absolute space] or across it, by an amount depending in the square of the ration of their velocities to that of light".
Briefly, what Michelson and Morley did, was to measure the speed of light in different directions. It was supposed that a variation in the speed of light would occur depending on the orientation of their experimental apparatus relative to the ether. For just as boat has a different speeds relative to the land when it moves upstream, downstream, or across the stream, light was expected to have different speeds also, depending on the velocity of the ether past the Earth. No difference was detected at all. This was a great un-solvable puzzle for classical physicists who were living in the realms of Galilieo and Newton. A number of explanations were put forth, however, most of them led to contradictions and the others just didn't make sense.
Then in 1905, Albert Einstein proposed a radical new theory; a theory that would overthrow all the work Newton and Galileo had done on relativity. With the new Theory of Special Relativity, the 26-year old Einstein overthrew old conceptions that were accepted, and replaced them with his new conceptions of space and time that would ultimately change the way we view the universe we inhabit.
See also, General Relativity and Special Relativity.