The brightest stars in the sky of Earth are also the ones with the most ancient names. Names such as Betelgeuse, Achernar, Sirius, Deneb, and Algol are usually Arabic, dating from around the tenth century when Arab astronomy flourished.
Less-bright stars were usually not given proper names at the time, and were thus harder to talk about.
In 1603, a German lawyer by the name of J. Bayer codified a more comprehensive system of stellar nomenclature: take the genitive (posessive) name of a constellation, and start labelling the stars with Greek letters in order of descending brightness. The brightest star in Centaurus would be Alpha Centauri, the second brightest would be Beta Centauri, the third brightest would be Gamma Centauri, et cetera.
This system was not adhered to rigorously, though. Bayer labelled the stars in Ursa Major by their position in the Big Dipper, rather than by their relative brightness; thus, Delta Ursae Majoris is considerably dimmer than Epsilon, Zeta, or Eta Ursae Majoris.
Eventually, though, youíll run out of Greek letters — there are only 24 of them, after all. Furthermore, at the dimmest limits of naked-eye visibility, it becomes very difficult to rank stars by relative brightness. A later Astronomer Royal named John Flamsteed ignored Bayerís naming scheme and suggested numbering the stars going from west to east rather than in order of brightness, since itís kind of hard to distingush relative brightness among the dimmer stars. The westernmost star in Centaurus would be 1 Centauri, the star immediately to the east of that would be 2 Centauri, et cetera.
The astronomical community accepted his new numbering scheme, but retained Bayerís Greek-letter names for the brighter stars that already had them. Thus, many stars visible to the naked eye have an Arabic name (such as Deneb), a Bayer Greek-letter name (such as Alpha Cygni), and a Flamsteed numeric name (such as 50 Cygni).
The Flamsteed system worked just fine until astronomers started using telescopes. Then, suddenly, a whole slew of new stars showed up that were too dim to be seen with the naked eye. The west-to-east numbering system of Flamsteed was simply not up to the task. In fact, no system was up to the task. The stellar astronomical community exploded into a flurry of catalogs, with each astronomer listing newly discovered stars by his own numbering system. Some astronomers at least had the decency to number the stars according to what declination (degrees north or south of the celestial equator) they were discovered at, so that we get catalog names such as Bonner Durchmusterung (BD) +5°1668 and Catalogue Astrographigue (AC) -24°2833-183. However, most astronomers just numbered them in the order they discovered them, giving us such creative and informative star names as Henry Draper (HD) 95735. And, worse for the peruser of astronomical literature, almost all star catalogs overlapped each other, so that a star identified as 95735 in the Henry Draper catalog is the same object as number 21185 in the Lalande catalog.
The only way to pin down a given star, then, is to give its right-ascention and declination in a given year (or "epoch") out to the greatest accuracy possible, and use that as your reference point when looking into the various catalogs.
Note that variable stars have their own naming convention which, while being somewhat convoluted, is at least consistent across the astronomical community.