There are two main types of galaxies (among other types), elliptical and spiral. These are then further sub-categorised.
Elliptical galaxies have categories from E0 to E7. E0 is a near spherical galaxy whilst E7 is highly elongated.
Spiral galaxies are ones that have arms. They are categorised with a letter S followed by a, b or c, depending on the tightness of the spirals and the size of the central bulge.
Most spiral galaxies are known as barred spiral galaxies (denoted with SB instead of S). These galaxies have a linear band of stars that stretch out from either side of the core into the spiral arms. Our own galaxy comes under this classification.
Other unusual shapes include ring galaxies (which have a core surrounded by a ring of stars), and lenticular galaxies (which have properties of both elliptical and spiral galaxies).
Many galaxies that have been found are known as dwarf galaxies. These galaxies are about 1/100th the size of the Milky Way and only have a few billion stars.