Most of the brighter stars were assigned their first scientific names by the German astronomer Johann Bayer during the early 17th Century, in 1603, in his star atlas Uranometria (named after Urania, the Greek Muse of Astronomy, along with Uranus, the Greek god of the sky and heavens). Johann Bayer traveled by sailing ship to various parts of the world, including the southern hemisphere, to conduct his astronomical observations and apply his data. Bayer assigned a lower-case Greek letter, such as alpha (α), beta (β), gamma (γ), etc, to each star he cataloged. Bayer then attached to each star’s Greek letter the Latin name of the star’s parent constellation in genitive form (or possessive form) to indicate that the star belonged exclusively to that constellation. (See List of constellations for a list of constellations and the genitive forms of their names) For example, Bayer assigned the Greek letter alpha (α) to a specific star in the constellation Taurus (the Bull), and added to the star’s Greek letter (α) the name Tauri, which is the genitive form of the Latin name Taurus, to indicate that the star belongs exclusively to Taurus the Bull. As a result, the star’s scientific name turned out to be α Tauri (that is to say Alpha Tauri), which means "Alpha of Taurus" or "Alpha of the Bull". A single constellation may contain fifty or more stars, but there are only twenty-four letters of the Greek alphabet, so, when he ran out of Greek letters to use for identifying the stars of a specific constellation, Bayer began using lower-case Latin letters. For example, Bayer assigned three stars in the constellation Carina as s Carinae, and another star in Centaurus as d Centauri, to indicate "s of the Keel" and "d of the Centaur", respectively. Within constellations having an extremely large number of stars, Bayer eventually advanced to upper-case Latin letters, ending with the upper-case letter "Q". For example, Bayer assigned a star in Scorpius the Scorpion the scientific name G Scorpii, which means "G of Scorpius" or "G of the Scorpion". Another example is a star in Vela the Sails, called N Velorum, which means "N of Vela" or "N of the Sails".
Orion provides a good example of Bayer's method. (Remember that the lower the magnitude, the brighter the star. Additionally a "2nd-magnitude" star has a more precise magnitude between 1.51 and 2.50, inclusive.) Bayer first designated the two 1st-magnitude stars, Betelgeuse and Rigel, as Alpha and Beta, with Betelgeuse (the shoulder) coming ahead of Rigel (the foot), even though the latter is the brighter.
He then repeated the procedure for the stars of the 2nd-magnitude. As is evident from the map and chart, he again followed a "top-down" ("north-south") route.
The belt of Orion is composed of three bright stars Delta Orionis, Epsilon Orionis and Zeta Orionis, however, the brightest star in the belt is not delta but epsilon. Instead, Bayer named the stars of Orion's belt in the special order in which they rise in the east, first delta (δ), then the middle star epsilon (ε), then zeta (ζ).
Although the brightest star in Draco is Eltanin (Gamma Draconis), Thuban was assigned alpha (α) by Bayer because Thuban, in history, was once the North Star 4,000 years ago. Almost every star with a history of being the North Star, including Vega, Alderamin and Polaris, were designated as the alpha (α) of their parent constellations by Bayer.
Sometimes, indeed, there's no apparent order, as exemplified by the stars in Libra and Sagittarius, where Bayer assigned designations to stars at random. (The letters of the Greek alphabet were used in antiquity to represent the successive integers; so Bayer's scheme might be regarded as equivalent to a numbering system.)
A further complication is the use of numeric superscripts to distinguish between stars with the same Bayer letter. Usually these are double stars (mostly optical doubles rather than true binary stars), but there are some exceptions such as the chain of stars π1, π2, π3, π4, π5 and π6 Orionis.