Answer
Electric scooters, quads, go-peds, mini-motos, hoverboards and Segways
In legal terms, these are all examples of vehicles that may be considered motor vehicles and are therefore subject to all the usual legal requirements that apply to other motor vehicles such as cars or motorcycles.
Therefore, they cannot be used on the road unless they are taxed (if required), registered, have an MOT (if required), are insured and the driver has a valid driving licence for the category of vehicle. If any of these requirements are not met, the vehicle can't legally be driven/ridden on the road.
In reality, many of these types of vehicle will never be 'road legal' because their design fails to meet road-vehicle safety standards.
Note also that it's an offence to use such vehicles on the pavement too.
They cannot be used on council land e.g., parks, unless there is a designated area specifically for them. In some areas, there may be commercially operated parks where they can be used.
They can only legally be used on private land/property with the permission of the land/property owner. However, note that it's against the law for a child under 13 to drive/ride on a tractor or self-propelled vehicle e.g., a quad, when it's being used in agricultural operations.
Electric bikes
Electric bikes known as Electrically Assisted Pedal Cycles (EAPCs) can be used on the road as long as they meet certain requirements (see Q605) and the rider is at least 14 years old.
EAPCs aren't classed as motor vehicles and so don't require insurance and the driver doesn't need a driving licence. Additionally, they are exempt from having to be registered and vehicle tax.
EAPCs cannot be used on the pavement.
Electric scooter trials
In some parts of the country, you can legally rent an electric scooter as part of a government trial – this is explained in Q1062.