When we look at the sky, there are no names next to the stars; and no name we assign to a star truly belongs to it, or necessarily says anything about it as a real object. It is merely our way of saying "Look at this star over here, not that one over there."

Common, or "Proper" Star Names
     Over the course of history, most stars have been given names, to differentiate them. Unfortunately, the same name was occasionally used for different stars, and most stars have been given many names even by people living in one area, and of course completely different names in different parts of the world. As an example, the current Pole Star (or North Star) is called Polaris, in acknowledgement of that status. In ancient times, however, the Celestial Pole was closer to Kochab than to Polaris, and Polaris was called Kynosoura (the Dog's Tail), which was also the name of its constellation (now called the Little Bear, or Ursa Minor). Similarly, Vega, one of over forty names used for the brightest star in the constellation of Lyra, was once used as the name for that constellation.
     Some of the names which we still use for stars are the names of mythological figures, such as the hunter Orion, the strongman Hercules, and the mythological warriors Castor and Pollux. Others describe the star's position in the sky. As an example, the Sirius is based on a Greek word meaning 'scorching', because its heat was thought to add to that of the Sun, to produce the hot "dog days" of late summer. It is also called the 'Dog Star', because of its position in Canis Major, the Big Dog. The name of the bright star to its northeast, Procyon, is based on the Greek 'pro kion', or 'before the dog', referring to the fact that being further north than Sirius, it rises before it, despite being further to the east. (More to follow, ASAP)
     (For a more comprehensive look at the history and mythology of star and constellation names, you can hardly do better than Richard Hinckley Allen's Star Names: Their Lore and Meaning.)

Bayer Designations
     In 1603, while preparing a "modern" star map, Johann Bayer decided to avoid the confusion associated with having different names for different stars by using the letters of the Greek alphabet to create what we now call "Bayer designations". The brightest stars in a constellation were given letters at the beginning of the alphabet, and fainter stars were given letters further along the alphabet. Thus the brightest star would be a (alpha), the next brightest b (beta), and so on. He did not follow only this rule, however. When, as in the case of the Big Dipper, there are a number of stars of roughly equal brightness, he used their positions to decide which would come first, and which would come next. In Orion, which contains two first magnitude stars, Betelgeuse, which is on "top" (being one of Orion's shoulders), is designated a, and Rigel, which is on the "bottom" (being one of Orion's knees), is designated b, even though Rigel is about thirty percent brighter than Betelgeuse. Similarly, in the Big Dipper, the stars are labeled according to their position, with Dubhe, at one end of the bowl of the Dipper, being designated a, and the other stars being designated b, g (gamma), d (delta), e (epsilon), z (zeta) and h (eta) in order of position around the bowl and down the handle, even though one of them, Megrez, is substantially fainter than all the others.
     The Bayer designation does not consist of only the Greek letter assigned to the star. It also includes the possessive form of the constellation's Latin name. Dubhe is a Ursae Majoris, or "alpha of Ursa Major", while Betelgeuse is a Orionis, or "alpha of Orion". On maps, only the Greek letter is shown next to the star, because the constellation is inferred from the overall position in the sky, or indicated by an outline of the area covered by the constellation, and the name of the constellation, somewhere inside that outline. In tables, the Greek letter is combined with a three-letter abbreviation of the constellation name, so that Dubhe is a UMa, and Betelgeuse is a Ori. Similarly, Polaris is a UMi, or a Ursae Minoris, meaning that it is one of the brightest or most important stars in Ursa Minor, while Kochab, which is equally bright, but no longer serves as the Pole Star, is b UMi, or b Ursae Minoris.

Flamsteed Numbers
     In the late 1600's, John Flamsteed extended Bayer's idea, using numbers instead of letters (in a sense, Bayer was numbering them as well, since the letters of the Greek alphabet were used as counters, as well as letters). Instead of just numbering the brighter stars, as Bayer did, Flamsteed assigned numbers to all the naked-eye stars (that is, all the stars visible in a dark sky, without optical aid). However, instead of using low numbers for brighter stars and high numbers for fainter stars, which is essentially what Bayer was doing, Flamsteed numbered the stars according to their right ascension, so that the westernmost star in a constellation was assigned the number 1, the next star to the east of that in the same constellation was assigned the number 2, and so on.

(more to follow)

Variable Star Names
     

Catalog Numbers and Positional Designations