ENIAC is generally considered to be the world's first general-purpose computer. Its first programmers were known as the "ENIAC girls": Kathleen McNulty, Frances Bilas, Betty Jean Jennings, Elizabeth Snyder Holberton, Ruth Lichterman, and Marlyn Wescoff.

There were individuals who had thought about such things before (e.g. Ada Lovelace, as Teleology mentions), but to my knowledge these are the first people whose job it was to make programs run on a functioning general-purpose computer. Programming as a profession began with them.