The jet engines on small business jets are usually started by an electric starter/generator. This is an electric motor that spools up the engine to a certain speed so fuel can be injected and ignited by the igniter plugs. After startup the starter becomes a generator that supplies the aircraft with electrical power. A small jet can start itself with battery power, but it is hard on the batteries and an external power cart will be used if available.
Larger jets have engines that are too big to be spooled up with electric motors. They have an APU (auxiliary power unit) in the tail, which is just a small jet engine used to supply bleed air and electrical power to the aircraft for use on the ground when it is not economical to run the main engines and for back-up if the main systems fail. Often the APUs are just a modified version of an engine used on some business jets.
If the APU is inoperative it is usually legal to fly the aircraft, however it will be incapable of starting itself. A ground power cart that supplies compressed air will be necessary. Once one engine is started it is possible to cross bleed air to other engines to start them too, much the same way that fuel can be transfered from one engines tank to another using cross feed. Some of the older jets like DC-8s didn't have APUs and some of these had airbottles so they could start one engine without a power cart, then do the rest with bleed air.
It is often a misconception that bleed air is 'exhaust' from the jet engine. It is taken from the compressor prior to the combustion chamber. Often large engines have multiple places to tap bleed air from depending on the pressure required. You might hear something like '8th stage bleed air' this means it is taken from the 8th compressor stage. The same engine might also have '12th stage bleed air' that is taken later after the intake air is more compressed. Bleed air will be used for cabin pressurization, engine and wing deicing (it is compressed so it is very hot), and powering backup hydraulic pumps, etc.
Bleed air does reduce the power available from the engine however and is not 'free'. Since it is taken before the combustion chamber it reduces the thrust available. On very large aircraft this is probably not a factor, but higher engine settings are needed to take off when anti-icing is turned on. Some small jets have boots on the wings for deicing because of this (some Citations) although they have bleed air or electric anti-icing infront of the engines so the boots will not knock chunks of ice into the air intakes of the engines and damaging the turbine blades. Others like the Hawkers use TKS weeping wings, so they don't have to use bleed air.