This powerpoint goes through the mechanics of a Michelson Interferometer as well as the theory behind how one works. There is a brief mention of an application of the interferometer, the Laser Interferometer Gravitational-Wave Observatory (LIGO).

1. The Michelson
Interferometer
oundracer.deviantart.com/art/Waves-wallpaper-118672151

2. The Mechanics: Visualized

3. Why can we see an interference pattern?
When the distance between the
reflected beam and the beam that
passes through mirror 1 is the same, the
waves are perfectly in phase:
But when the distance is slightly
off, for instance, by a half of a
wavelength, the two light waves
are perfectly out of phase:

5. How about the rings?
Away from the very very centre of the light beam, the distance of the light
beam is increased slightly at the edges of the beam (because of a slight angle
from the source), resulting in an in-and-out of phase behaviour from the
edges of the beam:
Still confused? Think of when you use a flashlight—the very edges of the light beam
usually fades off into the dark. This is analogous to the edges of the light beam from the
interferometer—it’s just at a much smaller scale.
//Note that other than the rays at the
very very centre of the beam, the
other rays are at a slight angle
(not perfectly perpendicular to the
receiving surface).

6. How can we use this?
An application of this actually magnifies this interferometer to a much much
larger scale—in fact, 4 km long “arms”, or distances between the mirrors and
the beam splitter (mirror 2).
Update/Bummer alert: They haven’t found anything yet from the first North American
observatory, from 2002-2010. They’ve built a new one in Hanford, Washington that’s expected
to be much better/sensitive.
Source: https://www.youtube.com/watch?v=RzZgFKoIfQI
Einstein proposed that gravitational
waves result from the collision of two
great masses, and the Laser
Interferometer Gravitational-Wave
Observatory was built in search of the
evidence of gravitational waves. Basically,
the fact that the arms are so far apart are
supposed to detect very, very, very slight
changes in distance as the result of pulling
and stretching from gravitational waves.
https://www.ligo-wa.caltech.edu/