Devised by amateur astronomer Johann Bayer in 1603, the Bayer Designation is probably the best known system of stellar naming and designation. In this system, stars are identified by Greek letters, followed by the genitive form of the constellation's name (the genitive form is a grammatical form in Latin that is equivalent to the possessive case in English). The Greek letters usually denote the order of brightness, so the alpha star is usually the brightest in a constellation. Thus, we call the brightest "star" in the Centaurus constellation Alpha Centauri (since it was first named, Alpha Centauri has been discovered to be a three star system).
In order to keep the naming system from getting too awkward, each constellation has a three letter designation. For example, Ursa Major is UMa. Thus in the Bayer system we would refer to the brightest star in Ursa Major as "αUMa", the next brightest one as "βUMa", and so on.
Sources: The Electronic Sky (http://www.glyphweb.com/esky/), Hyperspheres, Hyperspace, and the Fourth Spatial Dimension (http://www.cyburban.com/~mrf/)
Also, a quick note: some browsers might not display the α (alpha) and β (beta) characters in the last paragraph correctly. If you see a jumbled mush, that's why. I apologize in advance.