Maxwell's equations predict that electromagnetic waves, or light, travels through the aether at a constant velocity. However, this is only true from the aether's perspective; meaning from a viewpoint at which the aether isn't moving. However, if this is true, then from Earth's perspective, the aether is moving through Earth. In physics this is called relative motion, and the shifting in perspectives is called shifting  frames of  reference.

To understand this,  look at the following animations:

From the Aether's perspective the earth is moving through space while the aether is stationary.

From Earth's perspective the aether is moving through Earth while the Earth is stationary.

Shifting from one frame of reference to the other.

This is important because light moves at the speed of light only from the stationary aether frame of reference:

If the aether is moving relative to the light source, then the light will travel at different speeds in different directions:

The red rays are the light from the aethers perspective. The center of the red rays moves with the aether.
The blue rays are the rays from the source perspective. They move at a different speed, depending on the direction they move relative to the aether's velocity.

This gives us a property that can be measured - the relative velocity of the Earth and the aether.

Albert Michelson and Edward Morley in the late 19th century thought of  an experiment that would measure this - the relative velocity of the Earth and the aether.