The next step on the road to relativity was the development of the equations of electromagnetism by James Clark Maxwell. One direct consequence of these equations was the prediction of electromagnetic waves, which were quickly identified with things such as light.
One of the first difficulties of electromagnetic waves was the apparent lack of medium. Unlike other wave phenomena, like waves on a string or ocean waves, physicists knew little of what was "waving" in the case of an electromagnetic wave. To get around this difficulty, they just assumed that there was a medium filling all of space called the luminiferous aether, in which electromagnetic waves propagate.
A second difficulty was that Maxwell's equations predicted that these electromagnetic waves always traveled at the same speed c, regardless of the speed of the observer. This immediately led to the question "With respect to what?" Of course, the accepted answer at the time was with respect to the rest frame of the luminiferous aether, called the aether frame. This was an attempt to salvage Galilean relativity, which requires the speed of the wave to depend on the observer.
The next advance came from people attempting to study the aether.