Bayer designation is a classification system for stars after their brightness where each has a Greek letter in lower case followed by the genitive form in Latin of the name of its constellation. The brightest star has the first letter in the Greek alphabet, the second brightest the second letter, etc, and the stars are always designated by the brightness as it can be seen from Earth (with the help of telescopes, not by the eye). Bayer designations can be written in three ways. So the brightest star in the constellation of Orion Betelgeuse is written either as α Orionis, Alpha Orionis, or α Ori where the genitive is abbreviated. Every constellation has countless stars, so when the Greek letters run out the Latin letters, also in lower case, are used.
The word "Bayer designation" is named after the astronomer Johann Bayer. He gave most of the brighter stars their first Bayer designations in his star atlas Uranometria from 1603.