This is a hard question to answer definitively because the terms and often misused or confused.
Often, an emulator is a complete re-implementation of a particular device or platform. The emulator acts exactly like the real device would. For example, a NES emulator implements the CPU, the sound chip, the video output, the controller signals, etc. The unmodified code from a NES castridge can be dumped and then the resulting image can be loaded into our emulator and played.
A simulator is a partial implementation of a device/platform, it does just enough for its own purposes. For example, the iPhone Simulator runs an "iPhone app" that has been specifically compiled to target x86 and the Cocoa API rather than the real device's ARM CPU and Cocoa Touch API. However, the binary that we run in the simulator would not work on the real device.