Spiral
Spiral galaxies are characterized by a large, flat disk made up of stars and interstellar material, with younger stars arranged in spiral arms or bars radiating outward in a spiral pattern, and by an ellipsoid-shaped bulge above and below the disk made up of older stars. The older stars in the bulge are often part of globular clusters, thousands of stars bound together by gravity. The Milky Way is a spiral galaxy containing 200 billion to 400 billion stars.
According to Nick Strobel's "Types of Galaxies" article on the Astronomy Notes website, spiral galaxies are distinguished by four specific traits: ordered rather than random rotational motion, interstellar material between the stars, new star formation in the disk and, lastly, their spiral shape.
Lenticular galaxies are a special type of spiral galaxy. They are characterized by a smooth disk without any spiral structure. This smoothness is due to the lack of new star production because of the exhaustion of the interstellar matter used to produce them. There are also several subclasses of spiral galaxies, subtypes a through d, that are based on their geometry.
Elliptical
Elliptical galaxies are in the form of an ellipse, a sort of elongated circle. They have very little rotational movement as a whole, and although the individual stars rotate around or orbit the centers of these galaxies, the statistical sum of their angular momentum is normally near zero. They have little or no interstellar matter and are very similar to the bulge from a spiral galaxy with the flat disk removed.
Irregular
Irregular galaxies are characterized by their lack of any definite structure, probably due to distortions by gravitational forces. They are usually small galaxies. According to Strobel, dwarf irregular galaxies are among the most common galaxies in the universe. (Dwarves are galaxies composed of a few billion stars.) Irregular galaxies may or may not be actively producing new stars.