Star designation
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The International Astronomical Union (IAU) is the internationally recognised authority for assigning designations to stars (and other celestial bodies). Many of the star names in use today were inherited from the time before the IAU existed. Other names, mainly for variable stars (including novae and supernovae), are being added all the time. Most stars, however, have no name and are referred to, if at all, by means of catalogue numbers. This article briefly surveys some of the methods used to designate stars.
Note that there are companies that purport to name obscure stars after paying customers, but these names are recognized by nobody except the registering company and the customer, and there is nothing to stop two companies from claiming the same star, or even one company from registering the same star to two customers. See the Buying a Star (http://enzerink.net/peter/astronomy/starfaq/index.html) FAQ for more information.
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Proper names
Many of the brighter, or otherwise interesting, stars have proper names. Most of these derive from Arabic, but there are also a few Latin ones, such as Polaris, and even a few English ones, such as Barnard's Star. See List of traditional star names for a list of some of these names.
Bayer letters
Johann Bayer introduced a system of designating the brightest stars in each constellation by means of Greek (or less often) Latin letters, a system which is still widely used. See Bayer designation for details.
Flamsteed numbers
John Flamsteed's numbering of stars within constellations have also remained popular, although Bayer's Greek letters are usually preferred when there is a choice. See Flamsteed designation for details.
Variable designations
Variable stars which do not have Bayer designations are given special designations which mark them out as variable stars. See variable star designation for details.
Catalogue numbers
In the absence of any better means of designating a star, catalogue numbers are generally used. A great many different star catalogues are used for this purpose, see star catalogues.de:Sternbenennung fr:Désignations stellaires it:Nomenclatura delle stelle