Though I couldn't find anything official from the government in a few minutes of searching, this site
seems to indicate that in Illinois, a moped is a type of motor driven cycle and has the following definition:
A motorized pedalcycle (moped) is a motor driven cycle. It produces two brake horsepower or less and has a maximum speed of 30 miles per hour. If the moped has an internal combustion engine, the displacement shall not exceed 50 cubic centimeters, and the power drive system shall not require the driver to shift gears.
I'm not sure what "brake horsepower" is (or how it is different from plain old "horsepower"), but it must be some measure of power. I've always heard motorcycle power quoted in CCs (cubic centimeters, measuring the displacement of the engine I believe?). I grew up in Michigan, and IIRC, the signs on the highway on-ramps specifically said something like "no motorcycles under 125 cubic centimeters".