Luminous ether was a theoretical substance that physicists of the 1800s predicted had to exist, so that they could explain the wave-like behavior of light, such as the interference patterns demonstrated by Young's double slit experiment.
Ether, Aether, and Other NamesThe luminous ether is often just referred to as "ether." This should not be confused with the chemical ether, which not only exists but should be handled with care. It is also sometimes called the luminiferous aether (if you feel like being particularly pretentious by throwing in unnecessary syllables and vowels).
What Was Waving
The vast majority of evidence in the 1800s led physicists to believe that light was a wave, but waves were understood as energetic vibrations of a medium. Since light could move through a vacuum, it was unclear what medium was vibrating to create the light wave. The luminous ether was put forward as the medium that is vibrating, but physicists still didn't know what it was.
Note: This is similar to the way the concept of dark energy is applied today. We don't know what it is, but it's put forward as the explanation to fit the data. Some physicists (and others) have put forward the idea that dark energy and the luminous ether are somehow the same thing, but these ideas have never been particularly well fleshed out.
The problem was that no one could actually detect the luminous ether. Attempts were made, most notably the Michelson-Morley experiment of the 1800s. All failed to detect a vibrating medium through which light traveled. When Albert Einstein developed his theory of relativity, which contradicted the luminous ether model and opened the door for wave-particle duality, the ether model was abandoned.