A star can have more than one recognised name allocated to it by different organisations or persons. If you take for example, Regulus and you look up the name at SimBad, the large astronomical database at Strasbourg University in France, you will see it has 51 different identifiers and name. I've detailed only some, the more well known designations. In addition to that, Regulus according to Wikipedia, has additional proper names including Cor Leonis, Aminous Basilicus, Lion’s Heart, Rex, Kalb al Asad, Kabeleced.
In 1603, German Astronomer Johann Bayer started cataloguing all the stars he could see. He used the Greek alphabet where the first and brightest star would be known as Alpha, so for example, Hamal was known as Alpha Arietis. For most constellations that was true, however for Gemini, the brightest star in the constellation, Pollux is Beta Geminorum. Alpha Geminorum is given to Castor. After the 24th star was Omega, Bayer used Latin Letters such 'A','B', etc. Once those letters were used, smaller letters were used after. N Herculis is an example of a later Bayer designation.
The list of stars was catalogued by the Bonn Observatory in Germany between 1859 and 1903. On of its finds was BD-12 5055 which is more widely known as UY Scuti, the current holder of the largest star so far discovered. The name of the star is derived from the position in declination of between -12 and -13 and the position it is in in the catalogue. UY Scuti is the 5055th star in the catalogue. Although Bonn Observatory discovered it, it wasn't until the early part of the twentieth century that its size was fully worked out. Had the size been calculated earlier, it might have been studied more in other star catalogues.
Stars with a Flamsteed Id (e.g. 2 Cygni) was given its designation by Astronomer Royal, Sir John Flamsteed (1646-1719). Although Flamsteed studied the stars and created the first major star catalogue, he didn't create the numbering system. Flamsteeds first catalogue called Catalogus Britannicus was published after Flamsteed had died. The numbering system was added by French astronomer Joseph Jérôme de Lalande (1732-1807) in a revised catalogue. Ref: Ian Ridpath
Gliese star catalogue was started by German astronomer Wilhelm Gliese who hoped to survey all the stars that within 25 parsecs (81.54 Light Years) from Earth. The Gliese was first published in 1957 and was limited to those only within 20 Parsecs of Earth. It was extended to 25 Parsecs by Richard van der Riet Woolley who updated it in 1970.
Those stars that are prefixed with GL are from the original catalogue and those that are prefixed with GJ are from Woolley's update. Some of the Gliese stars have a decimal number, these were inserted later on to as to keep the right ascension order. Wiki
Whilst many Bayer stars in the northern hemisphere, the equivalent system is the Gould Id. The Gould star catalogue came from American astronomer, Benjamin Apthorp Gould (1824-1896). It Gould who surveyed the southern hemisphere and labelled the stars. An example of a star is 105 G. Pup. Stars are characterised as having a number followed by a G then constellation name in short hand.
The Bright Star Catalogue is characterised by the prefixes HR and BS and YBS. HR is the more commonly used code for these stars nowadays. It details the stars that are of stellar magnitude 6.5 or brighter. Although in addition to stars, it also includes star clusters such as NGC 2808 (HR 3671). The latest version of the catalogue would seem to be in 1991 and be the fifth release. Ref: Harvard
Any star that has an identifier beginning with the letters HD or HDE (E meaning Extension) (e.g. HD 1061) was first analysed and recorded in the Henry Draper Star Catalogue. Although Henry Draper lived from 1836 to 1882, it was only after his death that the first catalogue was published in 1918. In all, 359,083 stars was recorded over numerous revisions of the catalogue from 1918 to 1949. The Catalogue got the name after Henry's wife agreed to finance the start of the project as long as it was known as the Henry Draper Memorial.
A star that has an identifier which starts with HIP (e.g. HIP60352) is one that was studied by the European Space Agency Hipparcos satellite from 1989 to 1993. During its run, the Hipparcos satellite studied the distance and features of stars such as their colour, spectral details, its motion. Not all stars proper motion was recorded by the satellite.
The satellite detailed over 118,000 stars in our galaxy. I have only detailed a tenth of the Hipparcos star database on this site. Regulus has an Hipparcos I.D. of 49669. HIP1 is a star in the constellation of Pisces if you was wondering. By my calculations, Hipparcos detailed 4200+ stars in the Centaurus constellation, the most of any constellation.
A Kepler star is a star that has been identified and examined by the Kepler Space Telescope as it searches for Exoplanets in an area of the sky by Cygnus , the swan constellation. The telescope is named after Johannes Kepler (1571-1630) was a German Astronomer who was the first person to describe the motions of planets around the Sun in a way that they could accurately be predicted. Ref: N.A.S.A
Many stars, in particular bright ones will have a common or traditional name such as Regulus in the constellation of Leo, the Lion. The I.A.U. (International Astronomical Union) only formally recognised traditional names in November 2016 Ref: I.A.U.
In addition to formally recognising traditional star name, they also gave other stars proper names such as Lich for example. The reason for Lich getting it a name for example is because it was the first star that was identified as having Extrasolar Planet(Exoplanet) in orbit round it. Lich is a pulsar and would bathe its planets in radiation so there's no chance of life on any of those planets.
Regor is one of the closest wolf-rayet stars whose name Regor is totally unofficial but is a lot easier to remember than its more traditional Bayer Designation. The star was named by American Astronaut Gus Grissom Jr, as a joke but it stuck amongst the astronomy community. Its a new name, its actually the name of another astronaut reversed, Roger.
There are a number of stars which start with a V and then a number. This name identifies them as being a variable star, a star whose size changes over time. An example of one of these stars is V2388 Ophiuchi, a Beta Lyrae type variable star. Variable star types are named after the first star of its type to be discovered.
In addition to the V name, they can also have a letter designation starting with the letter 'R' then once they get to Z, they start with double letters from RA, all the way up to RZ and then SS...SZ, etc etc. After using ZZ, they start with AA and then go all the way up to QZ, after QZ they use the V and a number system.
These stars don't sting and are not black and yellow as their name might imply. Wasp-12 is an example of one of these stars. The WASP (Wide Angle Search for Planets) project is an international collaboration searching for planets using the transit method. Unlike Kepler, they are fixed to one area of the sky and are able to search in all constellations. Their first WASP-1 was found in Andromeda.
SuperWasp is a predominantly British universities project utilising two large telescopes, one in the Canaries and the other in South Africa. Any planet that they discover, the new planet will get an identifier of WASP-nn where nn is a number. According to their official website, they have discovered over 100 planets using this form. Ref: Wasp Planets and SuperWASP.
Wolf-Rayet stars are stars that have processed all their hydrogen and are now fuses heavier objects. These starts have an identifier starting with the letters WR in addition to any other name official or unofficial (Regor) they may well have.
There are a few stars that are named after people, the first example that comes to mind is NML Cygni which is one of the largest stars so far discovered. It is named after the people who discovered it, Neugebauer, Martz, and Leighton in 1965. Other examples of stars named after people include Tabby's Star and Cayrel's Star both named after astronomers who studied them.
Despite what the advert from the star naming company might say, the name of the star will not be recognised by the I.A.U. or any other professional organisation. The IAU have a detailed page on the subject so it should answer any questions you might have before buying a star.
If you were able to name a star, you'd get a lot of jokers taking advantage of the opportunity and giving them hidden meanings. Regor is an example of where the name was created as a joke but it somehow along with other names he made up stuck.