It's All Greek to Me
Well, it's actually a mix of Latin, Greek and Arabic. As it turns out, most of the star names that we are familiar with today have their origins in ancient cultures.
About 1900 years ago the Egyptian astronomer Claudius Ptolemy (who was born under, and lived during, Roman rule of Egypt) wrote the Almagest. This manuscript was a Greek text that recorded the names of stars as they had been named by various cultures (most were recorded in Greek, but others in Latin as per their origin).
This text was translated by the Arab world into Arabic, and used by their scientific community. At the time, the Arab world was known for their keen astronomical charting and documentation. So it was their translation that became popular among future astronomers.
The names for stars that we are familiar with today, called their proper names (sometimes known as traditional, popular or common names) are the phonetic translations of their arabic names into english.
However, some stars, like Sirius, are still known by their Latin, or in this case, Greek, names. Typically this is found with the brightest stars in the sky.
Naming Stars Today
The art of giving stars proper names has been lost on modern culture. Occasionally a star is given a common name in honor of a astronomer or astronaut in light of their significant contribution to the field. However, in most cases stars are simply given a numerical descriptor to signify their position in the night sky.
This name is usually associated with a particular star catalogue. These catalogues group stars together by some particular property, or by the instrument that made the initial discovery of radiation from that star in a particular waveband.
While not as pleasing to the ear, these star naming conventions are useful when searching for a particular type of star in a particular region of the sky. Such is often the case when undertaking research.
Star Naming Companies
The last way in which stars are "named" is by means of a star naming company. Chances are that you have heard of this practice, or even participated yourself. You pay a small fee and you can have a star named after you or someone you love.
While nice, the problem is that these names are not actually recognized by any astronomical body. So unfortunately if something interesting is ever discovered about your star you aren't going to see it referred to by your name in the papers.
The primary, and universally recognized authority on star names (and pretty much all things astronomy) is the International Astronomical Union (IAU). And they to this day do not recognize names given to stars by outside companies.
And when a star is designated a proper name by the IAU they will usually assign it the name used for that object by ancient cultures if one is known to exist. Failing that, significant historical figures in astronomy are usually chosen to be honored. However, this is rarely the case either any more.