Cosmology identifies a taxonomy of supernovae based on the spectrum (and thus atomic content) of their ejected matter. Some types of supernovae only occur in certain types of galaxy, and others share non-spectral features in common with each other only. Despite this, it is important to recognize that these classifications depend on spectral activity and nothing else; it is still possible a supernova could have all the explosive features of one class and still be put in another based on its color. There are two main types, I and II, and several subclasses beneath each.
Type I Supernovae: All type I supernovae are classed so because they lack hydrogen (H) bands in absorption or emission when they are at maximum light. That is, the most common element in the universe is not present in these explosions.
Type Ia Supernova: Believed to originate from the explosion of a white dwarf star, these are distinguished by the silicon (Si) content of their ejected matter. Stars do not become white dwarves until they burn off all fusionable material, which explains the type Ia lack of H and the presence of the heavier Si. Type Ia supernovae occur in all types of galaxy: elliptical, spiral, and irregular. Because they are both bright and common, Ia supernovae have replaced cepheid variable stars for use as a standard candle.

Type Ib Supernova: These lack Si, and are rich in helium (He). In these stars hydrogen has been lost either through solar wind or interaction with another star in a binary star system. A type Ib supernova has never been seen in an elliptical galaxy.

Type Ic Supernova: Like a type Ib supernova, only the star's helium mantle has also been partially stripped away in whatever way its hydrogen mantle was. Type Ic supernovae are also not present in elliptical galaxies.

Type II Supernovae: All type II supernovae have H bands in their absorbtion or emission spectra. All of these occur only in the arms of spiral galaxies, and all are theorized to leave behind a neutron star after explosion.
Type IIb Supernova: He is dominant over H in the spectrum, though H is clearly present.

Type IIL Supernova: H is dominant over He, and the brightness drops off linearly after its peak. The L in the title is for linear.

Type IIP Supernova: H is dominant over He, and the brightness hits a three month (or so) plateau after its peak. It should be no shock that the capital P stands for plateau. SN 1987A is generally considered a type IIP, but a large faction of cosmologists contend that it was instead a IIpec.

Type IIn Supernova: These have weak or absent H absorption, and tend to be most measurable by their radio and X-ray emission. This classification was proposed in 1990 and is not universally accepted.

Type IIpec Supernova: This is a catch-all category for supernovae that have spectral hydrogen, but don't fit obey the properties of any of the above classifications. It was proposed in 1985 to take the place of three classes from the previous standard supernova classification.