In the US, it's a mishmash of state and federal laws.

The general outline is something like:

• Many states allow electric scooters with throttles, with varying power and speed limits, generally 15-20 MPH and 500-750 watts.
• The federal definition for what you can sell as a "low-speed electric bicycle" is, a motor "of" under 750 watts, it must have pedals, and (without pedaling) the motor can't let it exceed 20 MPH on level ground with a 170 pound rider.
• I think everyone's gotten around to legalizing the use of electric bicycles, but until recently, in some states, any electric bicycle was legally a moped or motorcycle.
• Most states follow the "class system", which looks like this: !!!! Class 1 is 750 W and 20 MPH maximum, without a throttle (pedal assist only). A lot of European electric pedal assist cycles have their speed limiter increased to 20 MPH for the US. In most states, this can be used basically anywhere a bicycle can be (except sidewalks if that's legal in the state), unless electric bicycles are explicitly banned. !!!! Class 2 adds a throttle to Class 1. In most states, this is banned on dirt bike trails by default unless explicitly allowed, but is allowed everywhere else a Class 1 is by default. !!!!* Class 3 increases the Class 1 speed limit to 28 MPH (except in New York, where they are IIRC only allowed to be used in NYC, and because of a 25 MPH car speed limit, are limited to 25 MPH). This is basically the European L1e-B pedelecs, and was created specifically for those. They generally are allowed to access roads, and bicycle lanes on roads, but not separated bicycle infrastructure unless it's explicitly allowed.
• Some states just follow the federal definition for what's allowed.
• Anything outside of this is generally a moped or a motorcycle, regulated exactly like an ICE motorcycle (which varies widely by state - common limits for mopeds are 30 MPH, 50 cc, 2 HP (1500 W), and an automatic transmission, though), and requires registration, usually requires licensing, and sometimes requires insurance.
• ...unless it's an off-highway motorcycle, which doesn't require licensing or insurance (but is also forbidden from use on the street), although some states may require it for accessing public ATV/dirt bike trails. !!!!* And then some states give a way to register an off-highway motorcycle for on-highway use, if you demonstrate that you've installed mirrors, turn signals, head/tail lights, and the like. (But, you need a license, registration, and insurance for that.)