Naked eye stars have many names, a legacy of history. Vega, the UBV "standard" and the fourth brightest star in the sky has over twenty names in regular use.
Star names are either of ancient origin, recent historical origin, or are assigned by astronomical organizations under the umbrella of the International Astronomical Union (IAU). No private organizations have the rights to sell or name stars.
Proper names like Sirius Procyon and Vega come from Arabic, Greek and latin.
Johannes Bayer, around 1600, assigned lower case Greek letter names to the stars more or less in order of brightness, rendering the brightest star in a constellation "Alpha," the second Beta," and so on. To the Greek letter name is appended the Latin possessive form of the constellation name. Thus the brightest star in Lyra, Vega, became Alpha Lyrae.
Note that the constellation name is always abbreviated with three letters, eg, Vega is α Lyr. The first letter is always capirolized. Constellations with two words such as Canis Major (Canis Majoris) have the first two of the three letters capitolized so Sirius becomes α CMa. The brightness rule is often violated as Bayer also factored in the position of the star within the constellation. The Greek alphabet has 24 characters so for fainter stars Bayer added with lower-case Roman letters, then upper case Roman letters. Non-Greek characters are rarely used nowadays like "h Persei" and "G Scorpii". And the actual number of names is increased by applying superscripts to stars that fall near one another (a string of stars in Orion became π1 Ori, π2 Ori, π3 Ori...).
John Flamsteed created a huge star catalogue in the eighteenth century. He listed the stars in order of right ascension within the constellation boundaries. For example, 1 Lyr is the Western-most star in Lyra, 2 Lyr the next... Vega is 3 Lyr.
In general, first magnitude stars are known by their proper names, Greek lower case letter names are used until they run out, and then the Flamsteed numbers are used. For fainter stars we use catalogue names (numbers). Constellation names are dropped and the stars are named according to position, generally right ascension, or angle to the east of the vernal equinox.
The "Bright Star Catalogue," which serially numbers 9000 naked-eye stars through sixth magnitude from west to east beginning at right ascension zero hours (zero degrees) for the year 1900. (Precession changes the right ascensions with time so getting the HR from a current RA can be tricky.) Now published at Yale, the BS catalogue derives from a catalogue produced at Harvard, so the star names take on the name "HR" for "Harvard Revised." Vega = α Lyr = 3 Lyr = HR 7001.
Faint stars need catalogue names too and the most famed general catalogue for fainter stars, the "Bonner Durchmusterung" (the Bonn Survey), was compiled in Germany in the nineteenth century and lists stars to tenth magnitude. It divides the sky into declination strips one degree wide and then serially numbers the stars from west to east according to the stars' right ascensions (for the year 1855). The catalogue name incorporates the declination. Vega, for example, is also "BD+38° 3238," which means the 3238th star in the declination strip between 38 and 39 degrees north. The BD covers stars from the north celestial pole down to -2 degrees declination. The rest of the southern hemisphere is covered by the "Cordoba Durchmusterung" (the Cordoba, Argentina, Survey), or CD. Canopus, α Car, HR 2326 is CD-52°914 (CPD-52 00914), or the 914th listed star between declination 52 and 53 degrees south. BD and CD are sometimes combined as "DM" for "Durchmusterung." Precession has now moved many stars out of their original declination strips.
The most commonly used catalogue for all stars is the Henry Draper (HD) Memorial Catalogue, which serially numbers stars through roughly tenth magnitude to the east of the vernal equinox (according to right ascension in the year 1900) independent of declination. The HD catalogue was created to list the spectral classes of over 300,000 stars. Vega is HD 172167, Canopus HD 45348.
In the 1960s over ten positional catalogues were combined into the Smithsonian Astrophysical Observatory (SAO) star catalogue. It serially numbers over 250,000 stars according to right ascensions (1900) to ninth magnitude in 10 degree declination strips from north to south, each strip picking up where the last one left off. Vega is SAO 067174, Canopus SAO 234480.
Nowadays the Hipparcos (Input) Catalogue is used whenever possible. The Hipparcos satellite was launched in the 1990s to measure precise distances to nearby stars. Over 100,000 stars are listed. Vega is HIC 91262, Canopus HIC 30438. In other catalogues, fainter stars (or astronomical objects such as galaxies, quasars, radio sources) are named only by coordinates (right ascension and declination) for a specific year. The Hubble Space Telescope Guide Star Catalogue, for example, lists 19 million stars between 6th and 16th magnitude.
When the stars are close to each other, a system of Roman letters generally applied in order of discovery or of descending brightness is used. The principal component is "A," the next brightest "B," and so on. Sirius ("Sirius A") is accompanied by a faint white dwarf companion "Sirius B".
Many stars vary in brightness. The first found, among them Mira and Algol, were bright and had proper or Greek letter names. But as more were discovered, astronomers needed a systematic naming system. Since Bayer's last Roman letter in any constellation was "Q," they adopted "R" for the first variable found that did not already have a name, to which was appended the Latin possessive of the constellation name. The first such variable known in Sagittarius is thus R Sagittarii (R Sag) , the first in Aquila is R Aquilae (R Aql). The sequence continued with new discoveries to Z. Finding more, astronomers went back up the alphabet to use double letters, beginning with RR, then going to RS, RT...RZ, then to SS, ST...SZ, TT, TU...and finally down to ZZ. To accommodate increasing numbers, the system then went to the top of the alphabet, AA, AB...AZ, BB, BC...BZ, CC...and so on to QQ...QZ, with J left out to avoid confusion with I. After 334 letter combinations, astronomers give up and just used "V" followed by a number, V335 Sgr following QZ Sgr.