Aether was the hypothetical material that fills the region of space. It was assumed to be the medium that allows light and gravity to propagate in space. Throughout the late 1800s and early 1900s, there were experiments that attempted to prove if the aether actually exists.
Flat-Earthers (and geocentrists alike) often use the results of these experiments in order to support their case that the Earth is stationary. But they are wrong. These experiments were conducted to prove if the Aether theory, or if one of its competing hypotheses —like the Special Relativity— better explains reality.