The Naming of Stars

Astronomers use a number of conventions to name stars. Most conventions have their origins hundreds of years ago. All star names and designations must be approved by the International Astronomical Union. The Centre de Données astronomiques de Strasbourg (CDS) provides a Dictionary of Nomenclature of Celestial Objects.

The following are some standard nomenclatures for stars.

Arabic names, e.g., Aldebaran or Betelgeuse, have a lot of lore behind them. You can learn about their origins and meanings in Allen's "Star Names and their Meanings".
Bayer designations order the stars in a given constellations from the brightest on down. The stars are in (Greek) alphabetical order. The brightest stars in the constellation Orion are Ori, also known as Betegeuse. The second brightest is Ori, or Rigel.
Flamsteed designations are ordered by right ascension. These designations are numerical. The westernmost star in Taurus is 1 Tau, the next is 2 Tau, etc.

The arabic, Bayer, and Flamsteed designations exist only for the very brightest, visible. stars in the sky. Most stars are known by catalog names. Common star catalogs include:

the Yale Catalog of Bright Stars. Stars in this catalog are designated as HRxxxx or BSxxxx. There are 9110 entries in this catalog, which includes most stars brighter than visual magnitude 6.5. This catalog contains much supplementary information about these stars.
The Henry Draper Catalog (HDxxxxxx) contains positions, spectral types, and magnitudes of about 225,000 stars (plus more in supplements) to about ninth magnitude.
The Durchmusterung catalogs (BD, CoD, CPD), (DMsdd xxxx, where sdd is the declination and xxxx is the running number in that declination band, beginning at 0 hours right ascension).
The SAO catalog (SAOxxxxxx) is a positional catalog of about 250,000 stars to about tenth magnitude.
The Hubble Space Telescope Guide Star Catalog (GSCxxxx yyyy) contains positions of stars to magnitude 16.

Most bright stars have many names. Betelgeuse is also known as Ori, 58 Ori, BD+7 1055, HR2061, HD39801, and SAO113271. There are other names too, including entries in double star and variable star catalogs.

Variable Stars

Variable stars are named by constellation, and in the order that they are identified. The first variable star in a constellation is called R (e.g., R Mon), the second S, and so on through Z. That works for 9 stars. The next set uses two letter designations, beginning with RR, followed by RS through RZ, then SS through SZ and so on to ZZ. Then one goes to AA ... AZ, BB ... BZ ... QZ. This can account for 334 variable stars in a constellation. Since there are often many more than this, variable stars after QZ are designated as V followed by the number of variables. The variable star after QZ Tau is V335 Tau. There are over 2000 variable stars catalogued in Sagittarius alone!

Bright stars, even if variable, do not get variable star names. The eclipsing variable Algol goes by its Bayer designation, Per, in the Catalog of Variable Stars.