The Bayer Designation for stars was developed by astronomer Johann Bayer for his 1600s Astronomical Catalog. It groups stars by constellation, indicating its individual stars with letters of the Greek alphabet, and indicating the constellation by following the letter with the constellation's name in Latin genitive form. For example, "α Orionis" would be for some star in Orion. Bayer's assigned designations remain in use.
Bayer generally labeled the stars in the order of Apparent Magnitude, the brightest being α, the next, β, but the designations do not always follow this exactly. An exact ordering would imply that the brightest star in Orion would be "α Orionis", but this designation is given to Betelgeuse, a bright star in Orion, but Rigel (designated "β Orionis") is brighter. Examples: Alpha Centauri, Beta Centauri, Epsilon Eridani, Epsilon Indi.
To include additional stars in a constellation, Bayer followed the Greek letter labeling with Latin letters, and subsequent astronomers have enhanced the system to use pairs of letters (I find seemingly-conflicting descriptions of the ordering rules, so I haven't described them). These days, only brighter stars with Greek letters are much referred to by these designations. References often spell out the Greek letter, e.g., "Alpha" for "α". Also, the constellations are sometimes abbreviated. Thus the following equivalent designations:
I've heard of references using the upper case Greek letter, i.e., "Α Ori" for α Ori, "Α" being upper case "α". This would be ambiguous if Latin-letter designations were still employed for additional stars.
In some cases, a superscripted number (2, 3, ...) on the Greek letter is used to designate another star, generally very near within the Celestial Sphere, e.g., Mu Octantis and Mu² Octantis.
Extra-Solar Planets are often designated by a designator for the star followed by b, c, d, etc., for the first, second, and third "object of lesser brightness" discovered. For an example in which the star is known by its Bayer designation, "Upsilon Andromedae b" designates the first planet discovered that orbits the star "Upsilon Andromedae".