In various continuities, Eggman has been known by several names. In the Japanese games, he has always been known simply as "Dr. Eggman". In all games released in America and Europe up to Sonic Adventure, however, he was known as "Dr. Robotnik."
Both names slipped occasionally in all regions. For example, Sonic the Hedgehog 2's Wing Fortress Zone and Sonic Drift 2 only had Eggman, Sonic Triple Trouble had Robotnik Winter Zone, and Sonic Jam had both names, as his profile was Robotnik and his statue next to Sonic said Eggman on it. On one episode of "The Adventures of Sonic the Hedgehog" Tails said "Yeah right EGGMAN!" to Robotnik.
In Sonic Adventure, however, Sega began a process of getting the American and European gaming communities used to the Japanese name, Dr. Eggman. Initially, both names were used, and he referred to himself as Robotnik while others called him Eggman. In the sequel, Sonic Adventure 2, when he is announcing that he is taking over the world, he refers himself as Dr. Eggman, but in the background were the words "Robotnik Empire". In later games, and also in the English dub of Sonic X, the name Robotnik was dropped almost entirely, though it can still be found on some promotional material and in the French Sonic X dubbing. In the Japanese version of Sonic X his eyecatch profile card gives his name as: "ROBOTNIK (Dr. Eggman)". Also, most airboards in the game Sonic Riders were built by Robotnik Corp.
General consensus now is that his real name is Ivo Robotnik (although the name "Ivo" is debatable) and his nickname or title is Eggman. This is supported by the surname of his grandfather, Gerald Robotnik and his cousin, Maria Robotnik.