Within the EU it is regulated. We have one class which is often referred to as “Pedelec”. These electric powered bicycles make at least 90 % of all electric bikes. They have an electric power assist, which means that they will only be powered as long as the user is cycling himself. This power assistance works up to 25 kph (15,5 mph). That means that you can ride as fast as you want, on your own power, but electric assistance works only as long as you cycle and only up to 25 kph. These bikes are treated as bicycles which means that they can travel everywhere where bicycles are allowed. They do not need any registration or insurance, and you are not required to wear a helmet.
The next class ist the so called “Mofa” (motorized bicycle). They are very similar to the pedelecs, but they can run on electricity alone, so you can stop cycling and let the motor do the job. They are limited to 25 kph as well, they require an insurance and a license plate (given by the insurance company). Wearing a helmet is mandatory. Minimum age of rider: 15 years. You need a license which is very easy to obtain.
Then there are so called speed bikes, which meet the requirements of tiny 50 cc motorscooters: maximum speed of 45 kph (28 mph), insurance plate and helmet mandatory. In addition you need a drivers license which in most cases is included in a car drivers license. Minimum age of rider is 16 years.
The fact that all these types of electric bicycles do look basically the same leads to some strange situation. Often people who ride a speedbike, use bicycle lanes to escape traffic jams. This can be dangerous because usually a bicycle rarely travels at 45 kph.
*Disclaimer (all rules valid for Germany, maybe thetre are some minor differences in other EU countres)