How are galaxies classified?

Answer

If I understand correctly, you are looking for a certain substructure in galaxies, or our galaxy (the Milky Way galaxy) in particular.

We distinguish three substructures in our own system. There is the ‘hump’, and somewhat flattened spherical distribution of stars quite close to the center of our galaxy. We ourselves belong with the sun to a second component, the ‘disk’, a fairly flat circular disk of stars, gas and dust, which rotates about the galactic centre; the height of that disk is of the order of a few percent of its diameter. The third component is the ‘halo’, a spherical component around the center and bump, which is similar in size to the disk, but extends to the same height above the disk. Other spiral galaxies also have these three components, but their relative importance varies considerably.

There are also galaxies that are not spiral galaxies, because they do not have a disk. Elliptical galaxies can be seen as a big bump; there are small to very large. There are also small galaxies that are quite shapeless. Finally, there are spectacular, sometimes large, rather irregular galaxies, often explained as products of a collision between individual galaxies.

There is also some substructure within the three components. Most stars are born in groups, and these survive for some time as ‘star clusters’. In our system we distinguish ‘open star clusters’ in the disk and ‘globular clusters’ in the halo. The first are rather small star clusters (still thousands of stars…) with a somewhat irregular shape, the second contain hundreds of thousands of stars, and thus survive longer than the first.