In my state, it varies by type of e-bike as to who can ride and how fast, based on the class of bike. Click here for a list of laws by state.
A Class 1 eBike, or low-speed pedal-assisted electric bicycles, is equipped with a motor that provides assistance only when the rider is pedaling and that stops providing assistance when the bicycle reaches 20 mph. These e-bikes are legal on any paved surface that a regular bike is allowed to operate.
Class 2 eBikes, or low-speed throttle-assisted electric bicycle, are equipped with motors that can exclusively propel the bicycle, but that cannot provide assistance when the bike reaches 20 mph. These e-bikes are legal on any paved surface that a regular bike is allowed to operate.
A Class 3 eBike, or speed pedal-assisted electric bicycle, is equipped with a motor that provides assistance only when the rider is pedaling and stops providing assistance when the bicycle reaches 28 mph. Operators of Class 3 e-bikes must be 16 or older and wear a helmet. Class 3 e-bikes are prohibited from Class I multi-use bike paths unless specifically authorized by a local ordinance.
Below is a simple visual infographic for determining what class your eBike falls into (from currentbikes.com)