Galaxies are large groups of stars that are spinning through space. Galaxies can be classified by their shape, groups, and many other attributes. The classification has five main types: ellipticals, the normal spirals, lenticular, barreled spirals, and irregular. There are also subtypes within these groups. The elliptical group contains types E0 through E7. Lenticular is type S0. Normal spirals have types Sa, Sb, and Sc. The barreled spirals contain SBa, SBb, and SBc. Irregular is just a broad category without any subtypes. Elliptical galaxies are just what they are called: ellipses. The different subtypes correspond to different axis ratios (eccentricities) and nuclei sizes. There are also unusual types of active galaxies, known as the Seyfert Galaxies, which have a very bright nucleus with dim spiral arms, and the Twin Lobes, which are part of Centaurus A.
Quasars used to be mistaken for galaxies. Quasars are interstellar objects that radiate an enormous amount of light. However, the light coming from quasars tends to fluctuate in brightness over very short periods of time. This shows that quasars are not very big, because it takes light years to travel through most galaxies. (The faster it fluctuates, the smaller the object must be.) Quasars have the ability to emit the light equivalent to that produced by a huge galaxy, but only be the size of a solar system such as ours. Scientists now think that quasars are actually extremely large black holes. The first two quasars discovered were 3C48 and 3C273. A strange phenomena occurs when our view of a quasar is warped by a gravitational lens.
There are also classifications according to star populations. There are two types of star population: Type I and Type II. Type I conatins irregular galaxies and the arms of spirals. These include blue giants and blue supergiants. Type II are the elliptical, lenticular and the nuclei of spirals. This type includes red giants, subgiants, and subdwarfs.
Galaxies are often observed with spectroscopes. They emit light, which can be analyzed to detect the presence of elements within the stars that a galaxy comprises. The spectral types are as follows: B (blue), A (white), F (yellow), G (orange), and K (red).
Galaxies are often categorized as belonging to large groups. They can be classified as pairs, triplets, and groups. Some examples are:
To give you an idea of the extreme size of galaxies, the Local Group is about two million light years across, and it is only a part of the Local SuperCluster!
Astronomy Plus CCD Images of Galaxies - images
Galaxies - colorful guide at Cambridge
Clusters of Galaxies - images and text info