For those people who are finding it difficult to understand the workings of the actual Michelson-Morley apparatus, allow me to offer this layman's analogy.
First recall that the Michelson-Morley experiment was designed to reveal the motion of the Earth through something called the Luminiferous Aether.
Toward the end of the 19th and beginning of the 20th centuries, a theory was coming into acceptance that light was a wave. Up till then light was conceived of as a stream of particles. That's what Newton thought.
This is a good place to summarize what standard Newtonian Physics says about a wave. To quote from Wikipedia, "In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities." Not the best definition, but it gets the point across. That definition covers mechanical and electromagnetic waves plus some other phenomena that benefit from the same math. But electromagnetic waves were yet to be discovered at the time of the Michelson-Morley experiment. So, calling light a wave in Michelson's time meant believing it was a mechanical disturbance. And a mechanical disturbance requires a medium to disturb. Water waves require water or some type of fluid (and an air-fluid interface). Sound waves work within air, fluids or solids. Of importance here is that a mechanical wave cannot propagate, or even exist, in a vacuum. But outer space, through which light passes to get to us from the distant stars and through which Earth itself moves, seemed to be a vacuum. Even worse, Newtonian Mechanics, which was being used to describe the motion of the planets and stars so well assumed outer space to be a vacuum (or close to it). And now the new definition of light as a mechanical wave required that outer space was not a vacuum. And if outer space was not a vacuum, not empty, then, whatever was in it should effect astronomical measurements. But, outside of being the support for light waves, it didn't. Readers today are used to thinking of a universe without an Aether. They don't realize the devastating consequences to Newtonian Mechanics that the existence of such an unmeasurable medium would have.
In Michelson's time, a lot of effort was being put into proposing measurable properties for this light carrying medium they called the Luminiferous Aether. And those properties made scientists very uncomfortable. Take mass density. The speed of a mechanical wave through a meduim varies with the mass density of the medium. Sound travels faster in water than in air and still faster in steel. Even though scientists did not know exactly how fast light traveled at the time they knew its speed was enormous. And such a speed would require the mass density of the Luminiferous Aether to be enormous, greater by far than the mass density of any planet. The universe was beginning to look like an invisible yet super dense block of concrete with the planets and stars like tiny air bubbles somehow moving through it. Think about calculating the gravitational attraction between the Earth and the Sun in such a universe. How do you take into account the millions of miles of concrete between them? How even could the planets and stars move? One might be able to accommodate movement by assuming zero friction between the Aether and the planets. But Newton's law of gravity would be unsalvageable. And that was only one of several hard-to-accept properties about the universe that the existence of a Luminiferous Aether required. It's important to note that the possibility of the Earth and other planets actually dragging part of the Aether around the Sun with them was not even considered. After all, Newton's force law says that forces are equal and opposite. So if a planet exerts a drag large enough to take part of this enormously massive Aether with it, then the Aether exerts the same drag on the planet. And that would mean the planet would quickly loose angular velocity and spiral into the Sun. In other words, the solar system we see today couldn't exist.
That was the chaotic scientific environment in which Michelson conceived of and built his experiment. He used a recent invention called an interferometer which, when properly set up, could produce interference patterns by splitting a stream of light from a source and then rejoining the two streams on a display screen (or directly on the retina of your eye). The fact that an interferometer could do that was considered one of the "proofs" that light was a wave. The realization that light was an electromagnetic wave and that electric and magnetic fields did not need a medium in which to exist and propagate, in my opinion, saved Newtonian Physics. Because that realization made the existence of the Luminiferous Aether unnecessary. But I am getting ahead of myself. The new belief in Michelson's day was that light was a mechanical wave.
Getting an inteference pattern in an interferometer at that time was no easy task. It required painstaking setup. And once set up, the slightest change to the apparatus or the environment around it could destroy the pattern.
This is a good place to explain what is required for two streams of light waves to create an interference pattern. The two streams must have the following properties:
1. Monochromatic:
All the waves in both streams must have the same wavelength (or close to it).
2. Same speed:
All the waves in both streams must have the same speed.
3. Polarized:
All the waves in both streams must be oscillating in the same plane.
4. Coherent:
The crests of both streams must consistently coincide at the detection surface, as also the troughs.
Another important point to keep in mind is that the setup of an interferometer does not require the two path lengths of light to be equal. As long as both paths were a whole number of wavelengths the interferometer created an interference pattern. One path length could be 1,000,000 wavelengths and the other 1,000,172. Whereas, if one path was 1,000,000 wavelengths and the other 1,000,172 + 1/2 wavelength no interference pattern was created. Since the wavelength of visible light runs between 400 and 700 nanometers, assuring an interference pattern using that light would require the total length of each interferometer path to be set to an accuracy of around 200 nanometers. And since each light path consisted of several pieces (see picture below) that accuracy would have to be cumulative over several measurements. Loosely speaking, each measurement would have to be accurate to about 30 nanometers. Fixing the distance between interferometer mirrors to that accuracy would not be a problem today. In Michelson's time it was not practical.
In fact Michelson knew he didn't need to make those measurements. There is another way to make sure that both light paths in the interferometer contained a whole number of wavelengths. Turn on the light and tweak the distance for one of the path lengths till an interference pattern appears. If you've seen pictures of his interferometer that's what the set screw on one of the mirrors is for. The multiple mirrors gave the light a longer path in a smaller package and a longer path gave the Aether more opportunity to drag the light sideways. Also, Michelson's method of polarizing the light source was quite primitive (though typical for his day). He cut a narrow slit in a disk and put it in front of the light source. Fortunately, it was known that light that reflected off a mirror became partly polarized. So, the slit plus the bouncing of the light off multiple mirrors gave him an acceptable amount of polarization. Nowadays you can buy an interferometer kit from a science project supplier that uses a pencil laser. The laser puts out a beam of light with a bandwidth far tighter than anything Michelson could ever hope for. And, laser beams are already highly polarized. So you don't need the slit or all the mirrors to do your own experiment.
A quick internet search turned up this interferometer kit for less that 300 US Dollars.
https://www.amazon.com/45-941-Description-Michelson-Interferometer-Kit/dp/B07RYQFTB1
This one looks impressive. It costs over 5000 US Dollars.
https://www.pasco.com/products/lab-apparatus/light-and-optics/advanced-optics/os-9258
Given the popularity of the experiment, I would bet that many universities have their own interferometer kits tucked away somewhere in their physics labs. A serious layman might be able to talk a physics department into letting them use it.
Figure 1. This figure illustrates the folded light path used in the Michelson-Morley interferometer that enabled a path length of 11m.
a is the light source, an oil lamp.
b is a beam splitter.
c is a compensating plate so that both the reflected and transmitted beams travel through the same amount of glass
(important since experiments were run with white light which has an extremely short coherence length requiring precise matching of optical path lengths
for fringes to be visible; monochromatic sodium light was used only for initial alignment).
d, d' and e are mirrors.
e' is a fine adjustment mirror.
f is a telescope.
Even with all the limitations described above, Michelson still thought he could use an interferometer to demonstrate the presence of the Luminiferous Aether. His argument went something like this. If the luminiferous aether exists and the interferometer (along with the Earth) is moving through it, the speed and direction of the interferometer through the Luminiferous Aether would effect the setup. That's what all those geometry calculations in the textbooks say. And once setup with an interference pattern, any change in the speed or direction of the interferometer through the Luminiferous Aether would almost certainly destroy that pattern. So Michelson setup an interferometer in his lab. Then, after the interference pattern had been achieved and recorded, he intended to rotate the apparatus. If the inteferometer was not moving through the Luminiferous Aether (or if the aether did not exist) the rotation would not make a difference. (Aether drag was not even considered at first.) But since the surface of the Earth was rotating and the Earth itself was orbiting the sun, the chances of the inteferometer being at rest with respect to the Luminiferous Aether was considered impossible. Also, even though the apparatus was designed to be rotated, the constant movement of the Earth meant Michelson didn't have to do the rotating himself. Everything on the surface of the Earth, including the interferometer was constantly changing its orientation/direction of movement with respect to the Luminiferous Aether. Regardless of how the apparatus was rotated, after rotation Michelson expected the interference pattern to disappear.
The fact that the interference pattern did not disappear was a surprising and serious challenge to the existence of a non-dragging Luminiferous Aether. Historical accounts indicate Michelson considered his experiment and the apparatus he built to be a failure.
Say Mr. Michelson is standing in the fog on the edge of a body of water and wants to find out if there is a current in it that would cause objects to get dragged downstream. He can't just throw something in the water and watch. He would not be able to see it through the fog. Let's say he has two identical boats. He also has two long poles not necessarily the same length. The important part of this analogy is that he doesn't have the ability to measure or match the lengths of the poles. He also doesn't have an accurate clock.
Here is how he decides if there is a current. He lays one pole at the water's edge parallel to the bank. He instructs Mr. Morley to drive one boat from one end of the pole to the other and back. He suspends the other pole over the water perpendicular to the first making two sides of a right triangle. He drives the second boat himself from one end of that pole to the other and back. Now, if both boats start together and both poles are the same length and there is no current, he expects both boats to arrive back together. If there is a current, the boat traveling parallel to the shore will gain some time during one leg of the trip but loose the same time during the other leg. So its trip will take the same time as if there were no current. Not so for the boat traveling crosscurrent. It will get dragged down stream by any current and will have to constantly steer a little upstream to get back, resulting in a longer distance to cover. That will take longer and the boats will not arrive back at the starting point at the same time.
But, since he did not accurately cut the poles he knows the poles are not precisely the same length. So the odds of the boats arriving back together, current or no current, are pretty much zero. What to do. Mr. Michelson asks Mr. Morley to be patient and keep re-running the experiment. With each run, Mr. Michelson cuts a small amount off the pole whose boat took longer to get back until he gets a run in which both boats arrive back at the starting point at the same time. This means he has one of two scenarios. There could be no current and both poles are now the same length. Or, there could be a current but the poles are now of such different lengths as to compensate for the drag effect of that current.
Now what? Here is the brilliance in the experiment. Mr. Michelson switches poles. Now, if there is no current and the poles are the same length, switching them will not make any difference. The boats will arrive back at the starting point at the same time again. But if there is a current and the poles are not the same length, switching them will screw up the compensation. The boats will not arrive back at the starting point at the same time. Mr. Michelson has a method of demonstrating that a current is making the paths of the two boats unequal without the use of clocks and without the use of a measuring tape.
You probably noticed that I have not mentioned Relativistic Physics at all. That is because the original Michelson-Morley experiment was conducted wholly within the same room. The light source, interferometer and observer were all at rest with respect to each other. If nothing is moving but the light, Newtonian Physics and Relativistic Physics will predict the same results. But "wait", you might say. "The Aether is moving. If the Aether existed and it was moving past the Earth, the two theories would predict different results for the experiment." Right. So here is a summary of the three possible conclusions about the results of the Michelson-Morley experiment.
There is no Aether:
The observed results are compatible with both Newtonian and Relativistic Physics.
There is an Aether but it is being dragged by the Earth:
The observed results are compatible with both Newtonian and Relativistic Physics.
There is an Aether and it is moving with respect to the Earth:
The observed results are not compatible with either Newtonian or Relativistic Physics.