Electricity is a natural force and can therefore not be invented, only discovered. Benjamin Franklin started looking into electricity and sort of discovered electricity's properties and origins when a lightning bolt struck his kite in a thunderstorm. Look it up to know more. Another key figure is Nicola Tesla.
Electricity was first invented by the ancient Greeks, because many years back, they established that rubbing fur on amber created energy. Benjamin Franklin also made a great contribution by establishing the connection between lightening and electricity.