This is primarily historical, and I'm sure one of the old timers that has been doing this longer than I've been alive will come in with a better answer.
We take for granted the on board debug capabilities and flash ROM in modern microcontrollers, DSPs and ASICs. In the old days, the processor may have either been one time programmable (use once), or require UV erasure before reprogramming (45 minutes under UV was a good rule of thumb). These types of parts do not work well for rapid iteration during development. In comes the emulator. Since most of these parts were DIP or LCC, they were usually socketed. An emulator board would easily plug into the processor socket or a connected header on the application hardware. The emulator emulated the processor that would ultimately end up there, and have full access to the hardware.
Parts that do not have on board debug facilities still exist and are in active production. The PIC18F14K50 comes to mind. If you want to debug it, you need a special debug board. Microchip has a family of them.