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 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." Given the vast numbers of stars scattered across the sky, such appellations are necessary to avoid total confusion; and there is something comforting about knowing what a given star or constellation is called, even if it is just a name established by some sort of historical convention.

     Imagine how difficult it would be to specify one particular star, out of the multitude found in even a small area of the sky, without any way of saying which star you are talking about. (Noel Carboni, apod060317)

Note: It has been brought to my attention that one of the references I used for the following is not to be trusted; as a result, a few of the statements below may not be correct. Until this note is removed, although the discussion below can still serve as an example of the confusion which can be associated with star names, it may be subject to revision.

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 many stars have been given different names by people living at different times (or of course, in different parts of the world). As an example, the current Pole Star (or North Star) is called Polaris, in acknowledgement of its status. But in ancient Greek (Hellenic) times the Celestial Pole was closer to Kochab than to Polaris, and Polaris was called Phoenice, borrowed from Ursa Phoenice, the Phoenician Bear, the name given to what we now call the Little Bear (or Ursa Minor) when the constellation was created out of parts of other constellations by Thales, about 600 B.C.E. Later, Ursa Phoenice became κυνόςουρα (Cynosūra), the Dog's Tail, and that name was used for the star for more than a millennium. It is only in the last few hundred years, as precession has brought the Celestial Pole closer and closer to Polaris, that it acquired its present name. Similarly, Vega, one of over forty names used for the brightest star in the constellation of Lyra, was once used as the name for the constellation itself.
     Some of the names which we use for stars and constellations are the names of mythological figures, such as Orion the hunter, Hercules the strong-man, and Castor and Pollux, The Twins. Others describe the star's position in the sky. As an example, Sirius is based on a Greek word meaning 'scorching', because its heat was thought to add to that of the Sun to produce the "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', because being further north than Sirius it rises before it, even though it is further to the east.

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 α (Alpha), the first letter of the alphabet, the next brightest β (Beta), the second letter of the alphabet, 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 α, and Rigel, which is on the "bottom" (being one of Orion's knees), is designated β, 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 α, and the other stars being designated β, γ (Gamma), δ (Delta), ε (Epsilon), ζ (Zeta) and η (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 α Ursae Majoris, or "alpha of Ursa Major", while Betelgeuse is α 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 α UMa, and Betelgeuse is α Ori. Similarly, Polaris is α UMi or α 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 β UMi or β 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) visible from Great Britain. However, instead of using low numbers for brighter stars and high numbers for fainter stars, which is what Bayer usually did, Flamsteed numbered the stars according to their right ascension, so that the westernmost naked-eye 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. As it happens, Flamsteed did not intend to use these numbers on a permanent basis, but only as a way of keeping track of things until he completed his life's work, the Historia Coelestis Britannica, a catalog of over 2500 stars; and when the work was finally posthumously published, it did not contain any Flamsteed numbers. In some ways this was reasonable, as the boundaries of the constellations were very vague in Flamsteed's time, and over time precession changes the positions of the stars relative to the grid of right ascension and declination, so that the order of the numbers does not exactly agree with modern coordinates.
     Of course, if Flamsteed's published work did not contain any of the Flamsteed numbers, the reader might wonder how we know what they are. That is due to the fact that over the decades spent on his work, astronomers anxious for an accurate catalog of the sky (not to mention the patron who was paying for Flamsteed's work) became more concerned about the long delay in its publication; and in 1712 Edmond Halley and Isaac Newton purloined a copy of Flamsteed's work and published 400 copies. Flamsteed was furious about this (not only because he had not authorized the publication, but also because he felt that being incomplete, it was an insult to the heavens themselves to publish a catalog that might contain a number of errors), and bought and publicly burnt 300 copies of the unauthorized publication, making the remaining copies true collector's items, especially since almost all of them belonged to famous astronomers. But since those remaining copies were used by the great catalogers of the sky (such as William and John Herschel) to designate which stars they used to determine the positions of other celestial bodies, the Flamsteed numbers have survived to this day, despite their originator refusing to acknowledge them.
     As it happens, one of Flamsteed's objects, 34 Tauri, represented an object that was not a star, but a planet. In 1690, more than 90 years before William Herschel's telescope revealed the existence of Uranus, Flamsteed observed Uranus six times, recording its nearly constant position as that of a star in Taurus. These were the earliest confirmed observations of Uranus, but it was observed as a supposed star on many other occasions before Herschel oberved it, and it has been suggested that even one of the stars in Hipparchus' 128 BCE star catalog may have been an observation of the planet. (Similarly, although Neptune was not discovered until 1846, it was observed by Galileo over 200 years earlier.)

(many more examples and methods of naming to follow)

Variable Star Names
     (far more to be added later, but basically, the first variable-brightness star discovered in Orion is called R Orionis, the second variable star discovered in Gemini is S Geminorum, the third variable star discovered in Auriga is T Aurigae, and so on. Once individual letters are all used up, double letters are used; so for example, we could have a star called RU Lupi in Lupus, or one called TV Ceti in Cetus, if enough variables were discovered in those constellations)

Catalog Numbers and Positional Designations
     (far more to be added later, but for instance there are objects in a catalog called the New General Catalog that are called NGC 1, NGC 2, NGC 3 and so on, objects in a catalog created by Charles Messier called M1, M2, M3 and so on, objects in a catalog created by Halton Arp called Arp 1, Arp 2, Arp 3 and so on, and on and on. And nowadays, with billions of stars and other objects in computerized catalogs, it is not unusual to simply specify an object by its position on a particular date, such as J2000 012345.6+012345, meaning the object with the position RA 01 23 45.6, Dec +01 23 45 according to the position of the Pole at the start of 2000 CE.)