The Eggman Empire, used interchangeably with Eggmanland, Robotnikland, and Robotnik Empire, is a concept which often crops up amidst the crazed ravings of Sonic's arch-nemesis and resident evil genius, Doctor Ivo "Eggman" Robotnik. Its meaning often seems to shift wildly depending on translation, continuity, and Eggman's mood at the time, but in general, the Eggman Empire refers to territory in Sonic's world that's been taken over by the fat scientist and his robotic servants.
Before Sonic Adventure began injecting a more concrete plot into the Sonic continuity, Eggman's precise motivation for all the robotization he kept committing was never entirely spelled out in-game. However, in the 3D era, the fat man's dreams of global empire became more blatant - notably in Sonic Adventure 2, when he declared as much to the entire world after firing the Eclipse Cannon. So this sense the Eggman Empire can also refer not to the territory Eggman has, but to the imagined territory he wants - an entire planet totally under his control.
Levels set inside into Eggman's factory-cities in the early games - Scrap Brain Zone, Metropolis - give a preview of what the Eggman Empire might be like. The area closest to Eggman's imagined dystopia seems to be Sonic Unleashed's Eggmanland stage; a crazed carnival-factory, chock full of egomaniacal statues depicting the scientist.