Well so basically the original thinking was that if electromagnetic waves travelled through a medium, then they would have a constant speed relative to this medium. As such, an object moving relative to this medium would measure different speeds for waves in different directions. No such effect was found, despite doing such things as measuring when the Earth was going in a different direction around the Sun.
These results decreased belief in an aether (though more complex experiments were needed to falsify the possibility that the Earth was dragging aether with it). Additionally, a series of length / time dilations were proposed for moving objects to explain the bizarre fact that the speed of the any light was measured the same regardless of the relative movement of the measuring observer. At this point Einstein stepped in and explained the theory of Special Relativity and how speeds don't add linearly and distances and times don't transform as one would naively expect under Galilean relativity. It was a really a generalization of the previously proposed Lorentz transformations that returned to the idea that all movement was relative. The spacetime that transforms in this way is called "Minkowskian".
Eventually after this, Quantum Mechanics developed into Quantum Field Theory, which is inherently relativistic and embedded in Minkowskian spacetime. In this view, electromagnetic photons are just excitations of the quantum field. Things basically came full circle and the quantum field is essentially an aether as envisioned all along. The difference is that it is relativistic / Minkowskian, and as such doesn't have a particular reference frame in which it is "not moving".
Note that even before the development fo QFT, this is what Einstein was saying: "More careful reflection teaches us however, that the special theory of relativity does not compel us to deny ether. We may assume the existence of an ether; only we must give up ascribing a definite state of motion to it, i.e. we must by abstraction take from it the last mechanical characteristic which Lorentz had still left it. We shall see later that this point of view, the conceivability of which I shall at once endeavour to make more intelligible by a somewhat halting comparison, is justified by the results of the general theory of relativity."
Einstein's whole letter is a great read: https://mathshistory.st-andrews.ac.uk/Extras/Einstein_ether/