The original system for naming variable stars was invented by Johannes Bayer near the turn of the 17th century. He designated variables by assigning a Greek letter followed by the Latin genitive case of the constellation name (ex. alpha Orion). When he ran out of Greek letters to use, he switched over to Roman letters (ex. A Orion). This scheme was sufficient for Bayer, as he never made it past the letter Q in any of his catalogs.
In the mid-1800's, a German astronomer by the name of Friedrich Wilhelm Argelander invented a new naming scheme, and it is still the one we use today. He followed Bayer's basic form, using letters followed by constellation name, but altered the form of the letter designation. At first, he used the letters R through Z but soon realized that he needed more letters for the numerous variable stars that existed in each constellation (there can be over 1500 variable stars per constellation!). So he began to double up the letters; RR-RZ, SS-SZ, TT-TZ, and so on. Still, these letters ran out! He decided to go back to A again; AA-AI:AK-AZ and so on up to QQ-QA:QK-QZ (J was not used for fear that it would be mistaken for I). Even these letters did not hold out, and astronomers started using numbers preceded by the letter V beginning with V 335 (the letter and letter pair combinations above total 334 combinations).
So, to summarize, variables are named: