The first country to introduce the adhesive postage stamp was Great Britain in 1840. From there, its use spread rapidly to both North and South America and to what is now the Commonwealth. Thanks to assiduous collectors, many stamps have survived in great numbers and most are worth pennies only. For a stamp to be rare, it usually has to be very early or to have a high denomination (its value when issued) or to be unusual because of some printing error or other circumstance.
Inspect your stamp for signs of age. Look for muted colouring, a slight tanning to the paper and softening to the corners. If a stamp is very early, it is usually "imperforate," that is, it lacks a perforated edge. Postmasters used to snip individual stamps from a sheet (known as a "block"). In the 1850s, a machine was introduced to punch lines of holes between the stamps.
Check the stamp's printed value or "denomination." The higher the denomination, the fewer would have originally been in circulation. Bear in mind that a printed value that seems inexpensive as of 2010 might have been quite costly to letter-writers of a hundred or so years ago. In 1893, for instance, a five-dollar stamp was issued to commemorate the Columbian Exposition -- a princely sum at the time.
Research the stamp in your local library in guides such as the "Scott U.S. Stamp Pocket Catalog." It's only by studying such guides carefully that you will discover if some other factor might come into play. Collectors place great store on subtle variations in paper or colour, tiny flaws in the printing process and what are known as "marginal markings," printer's codes which occasionally creep onto the stamps. For instance, there is a collectable variant of the 37-cent commemorative stamp for the Athens Olympic Games of 2004 with a marginal marking along its bottom edge. However, it can take some perseverance to spot these minuscule errors.
Examples of famous early stamps are the Brazilian 90 reis of 1843 (the so-called "Bull's Eye") and the Bavarian 1 kreutzer of 1849. The first widely available U.S. stamps were the 5- and 10-cent stamps of 1847, depicting Benjamin Franklin and George Washington respectively, and perforated stamps followed in 1857. Many 19th Century U.S. stamps are not real rarities, but they are scarce in comparison to the demand from collectors. Equally, some stamps can be very rare while commanding little commercial value. Stamps dating from after World War II are unlikely to be rare unless they are an unusual variant, but they are rich in social history. If you're interested in collecting flaws, then look out for little flashes of white and faint shadowing on certain lines -- unfortunately they're rarely obvious. Take a look at the 2-Clicks Stamps website for more help in this fascinating area.
Many reproductions of rare stamps have entered the market through the Internet. These are made with modern domestic colour printers, whereas, until the 1970s, U.S. stamps were recess-printed. That is, they were struck from copper plates under great pressure, a process which leaves tiny telltale ridges.