There are various categories of stamps. Here are some of them:
v Origin : issued countries
§ It refers to the place which issue stamps, e.g. USA, United Nations, PRC, Germany, etc.
§ It is usually the primary classification in the stamp catalogues, e.g. Scott, Stanley Gibbon, Michel, etc.
v Topic :
§ It refers to the theme of the stamp, e.g. animals, transport, Christmas, flag, New Year, Europa, etc.
§ The most common topics are the animals, Christmas, flowers, etc.
§ There are some stamp catalogues are specialised in some of these topics
v Format :
§ It refers to the presentation of the stamps.
Ø Single or Set: the stamps are issued in single stamps or in set
Ø Miniature sheet: the stamps are placed enclosed in a small sheet
Ø Souvenir sheet: a whole sheet consists of different stamps
Ø Se-tenant: a set of stamps attached to each other in a strip
Ø FDC (First Day Cover): the stamps are placed and stamped on a cover.
Ø Booklet: the stamps are placed inside a booklet
Ø Presentation Pack: the stamps are stored in a pack, which includes full description of the stamps
Ø Coil: the stamps are issued in a coil.
v Type :
§ It refers to the purpose of the stamps.
Ø Commemoratives: the stamps are issued for specific issues, e.g. Christmas, Anniversary, etc., which can be available at the office for short period of time.
Ø Definitive: the stamps are issued for mailing purpose, which can normally be available at the post offices for more than one year.
v Condition :
§ It refers to the physical conditions of the stamps.
Ø U/M (**) un-mounted mint: the stamp new and has not been mounted
Ø Hinged (*) the stamp has been hinged
Ø U the stamp has been used and stamped
Ø CTO the stamp has been stamped on its issue
Ø F Fine: the stamp is in fine condition
Ø VF Very Fine : the stamp is in very fine condition
Ø Gum the stamp has some dirt
v Shape :
§ It refers to the appearance of the stamps.
v Printing :
§ It refers to the printing method of the stamps.
Ø Texture the surface of stamp is uneven caused by the ink, usually old stamp of single colour, e.g. Bahamas, Fiji and other old British Commonwealth stamps.
Ø Flat the modern stamps with the flat and smooth surface
The above classifications are not comprehensive and are only based on my experiences.