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.
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.
[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 stars which do not have Bayer designations are given special designations which mark them out as variable stars. See [variable star designation]? for details.
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.