There are three main parts to a comet:
Nucleus - also known as the core of the comet. It is mostly made up of ice and dust, covered in other organic material. It is also the solid part of the comet.
Coma - the coma is a halo of gas that envelopes the nucleus. The gas formed is a product of sublimation. The coma and the nucleus combined form the head of a comet.
Tail - The tail is a stream of gas and dust that follows the comets head. Solar wind and solar radiation blow dust and gas from the coma of the comet which creates the tail. There are two main types of tails of a comet, the ion tail, and dust tail. (Sometimes a third tail may appear which is the plasma tail.)