Luminiferous Ether was supposed to be a massless vapor that scientists once believed filled the universe as the medium of transmission of electromagnetic waves, including visible light.
Here is a nice animated web site showing how a wave propagates in space: electromagnetic wave demo web site
Sound waves, however, can not travel through a vacuum. Sound in a gas consists of compression waves, formed when a gas (like air) gets compressed in a moving 'pulse'. Sound in solids can also be transverse waves, like the vibrations on a piano string. In a vacuum, there is nothing to be compressed or jogged sideways, so sound waves can not travel there. It was the analogy with mechanical sound waves that encouraged scientists to believe that a similar ether must be needed to propagate electromagnetic waves.
This kind of reasoning needed to be tested in nature to see if it was correct. If indeed electromagnetic waves travel through some kind of ether, then one might expect noticeable effects if we are moving through the ether at some speed. If electromagnetic waves travel in one direction they would travel faster relative to us, and in the other, they would travel slower, so that their speed relative to the ether is the same. This is what is observed with sound waves.
Michelson and Morley tested this hypothesis in 1887 by shining light in perpendicular directions to see if the speed depended on the direction of propagation. Their apparatus was mounted on a rotating table, and they did their experiments in the day and night, and at different times of the year, to see if our direction of motion through the ether would change. No evidence of an ether was observed -- the speed of light was observed to be constant for all directions and speeds of motion of the laboratory (taking advantage of the Earth's rotation and revolution around the sun).
This lack of evidence for ether and the validity of Maxwell's equations for describing electromagnetic waves in all reference frames led Einstein to propose in 1905** the special theory of relativity.
** According to my sources, Einstein made only passing reference to Michelson and Morley's experiment in his famous 1905 paper, and later could not recall if he was aware of the details of the experiment. Nonetheless, the validity of Maxwell's equations for electricity and magnetism in all reference frames is what Einstein set out to explain, and the experimental proof of that came from the groundbreaking work of Michelson and Morley.
An aside -- gravity is also predicted to propagate by waves in space as well, without the need for an ether. There is experimental evidence for this from rotating neutron stars slowing down in accordance with a loss of energy in the form of gravitational radiation.
(published on 10/22/2007)