nth generation; nth person to hold a position
Counter suffix attached to numbers to indicate the generation of a family business, dynasty, or the ordinal holder of a title or position. Often used for kabuki actors, sumo wrestlers, and company presidents.
See also: 代
He is the third-generation owner of this shop.
He is the fifth president (of the company).
That kabuki actor is the twelfth generation.
She became the fifth-generation president.
代 is a broader counter for generations, eras, or periods, while 代目 specifically counts the ordinal holder of a position or title, such as a kabuki actor or company president.
世 counts successive holders of a hereditary title such as a shogun or pope, while 代目 is the everyday counter used for things like company presidents, sumo wrestlers, or kabuki actors.
Compound of 代 (generation) and 目 (ordinal suffix).