Generally laws have been trending towards more acceptance of electric bicycles, although with codifying specific restrictions for their use.
The federal laws for what can be sold as a "low-speed electric bicycle" basically boil down to the following: 750 W maximum, must have pedals, and if the rider isn't pedaling, must not be able to exceed 20 mph on level ground.
That says nothing about what you're allowed to use, though.
For comparison, European laws for an electric pedal-assist cycle are: 250 W continuous maximum (note that continuous is after 30 seconds...), must be pedaled to get assistance (except for a 6 km/h (3.7 mph) walking assist mode), and no assistance above 25 km/h (15.5 mph).
There's also an S-Pedelec category in Europe, based on moped laws, that requires licensing and registration, and has reduced access to cycling infrastructure, but is allowed 500 W and 45 km/h (28.0 mph).
As far as what you can use... Some states allow anything that's legal under the federal law, some states allow nothing or classify it as some kind of motorcycle/moped (Ohio and New York come to mind), and then there's the California legislation that is being used as a model elsewhere (including in New York's bill).
The California law designates three kinds of e-bike. All three are 750 W maximum. The first is 20 mph maximum and must be pedaled to get assistance, the second is the same but can use a throttle - otherwise they have the same requirements (IIRC, nobody under 16 operating it, otherwise basically the same as a bicycle), but the class distinction is so that off-road trails can opt to exclude bikes with throttles. The third class is 28 mph maximum and must be pedaled (so an S-Pedelec falls into this class as-is), and has stricter age, helmet, and usage requirements (some cycling infrastructure is unavailable).
Worth noting that the bill in New York is only allowing the first class (which is essentially what NYC has effectively legalized).