Jet engines are usually started with an electric motor driven by batteries, or sometimes using pressurized air. when the first engine is started it gives the power required to start other engines(in fact the first engine to be started is Auxiliary Power Unit, that is a small engine used to provide power to the airplane) then the power provided by the engine started is used to start all engines.
Every engine on jetliners is equipped with alternators and pumps, so the engine provide electricity and hydraulic and pneumatic power (little airplanes usually have no hydraulic-power plant, only electrical plant.). But for example Tornado's main engines are started by the APU using a mechanical gear-box.
Batteries are needed when the power needed is more than the power produced, and if there is an engine's malfunction, and to start the APU.
Little planes work like cars, batteries, engine, alternator and the electrical start motor(sometimes there is only a device that acts for both)