The names of big groups of things in space go like this: Planetary system (planet, moons and rings), solar system, interstellar neighborhood, galaxy, local galactic group, supercluster, local superclusters, the universe.
A galaxy cluster is (as the name implies) a bunch of galaxies fairly close together (for galaxies), held together by gravity. Our galaxy is the Milky Way Galaxy, and our closest neighbor is the Andromeda Galaxy. Both are spiral galaxies, but spiral isn't the only shape a galaxy can come in. Some neat ones are sombrero galaxies and irregular galaxies.
Three major types of galaxies.
Our local galactic group.
“Spiral isn't the only shape a galaxy can come in.”
The formation of an irregular galaxy. Oh wait, that's toast. My bad.