can cars run on electricity instead of fuel  

Yes, hybribs are common in Canada and the US I have also see a car run off of Solar power and even plain H20 so it cane be done...

For the common automotive term of hybrid vehichle see: Petroleum electric hybrid vehicle
A hybrid vehicle (HV) is a vehicle that uses two distinct power sources such as:

An on-board rechargeable energy storage system (RESS) and a fueled power source for vehicle propulsion
Human powered bicycle with battery assist
A sail boat with electric power[1]
The term most commonly refers to petroleum electric hybrid vehicle, also called Hybrid-electric vehicle (HEV) which use internal combustion engines and electric batteries to power electric motors.

The term hybrid when used in relation with cars also has other uses. Prior to its modern meaning of hybrid propulsion, the word hybrid was used in the United States to mean a vehicle of mixed national origin; generally, a European car fitted with American mechanical components. This meaning has fallen out of use. In the import scene, hybrid was often used to describe an engine swap. Some have also referred to flexible-fuel vehicles as hybrids because they can use a mixture of different fuels — typically gasoline and ethanol alcohol fuel.

Yes of course. I know that inventors already invented car running in a electricity and I know that there is a car invented running in a solar electricity.

there are many models already available with electri model... the problem with them is they cannot take load... the best car in electric model can take not more than 2-3 people without heavy luggage... u can opt for an option which is not very common i.e. Alcohol based cars... they r cost effective and environmentally safe... problem with them is they do not have mass acceptance and the cost of the car is usually very high... u have to get conversion kits which r easily available in some European countries....

yes it is possible but i think that people witch earn by selling a fuel block this ...