Hybrids Electric Vehicles (HEVs or just Hybrids) weigh about the same as your average passenger vehicle and tend to look and drive very much the same. Some automakers have even created fuel-efficient Hybrid SUVs. Hybrids are not the same as the discontinued electric cars of some years back -- they are still largely powered by a standard internal combustion engine. Hybrids make better use of this engine with the additional of an electrical motor, the battery of which is charged through means that in a standard vehicle result in wasted energy -- braking, for example, or coasting. In a standard passenger vehicle, when you press the brakes the energy of the car's motion becomes heat as the friction between the brake pad and the disc causes the car to slow or stop. Hybrid technology harnesses this otherwise wasted energy to charge the electric motor; this motor then helps the engine during acceleration, hill climbing, and slower travel when the regular combustion engine is less efficient. This means that your car's engine needs less gas to do it's job, resulting in significantly increased gas mileage: up to 50 mpg in some vehicles!
Keep in mind that not all hybrid vehicles use their technology the same way; some, rather than use the electric motor to ease the burden on the combustion motor, use that energy to provide the car additional power rather than additional efficiency. If you're looking at purchasing a hybrid vehicle, begin researching the model at Wikipedia to see where it stands.