Most twin engine transport aircraft (IE 737, 777, A320, etc) really have 3 engines. The third engine is usually in the tail, and provides almost no thrust. The third engine is generally small (compared to the two engines on the wings) and is called the Auxiliary Power Unit (APU). This APU will provide the aircraft with electricity, air and maybe hydraulic power in the event the wing engines are unable to power the electrical needs of the aircraft.
The APU is a turbine engine, a small jet engine that will provide various resources, like a generator. The APU will use the fuel from the tanks on the aircraft. The engine runs at a constant speed, so there are no throttle controls. There are various gauges to monitor the performance of the engine and the accessories.
The APU can generate enough power to start the larger engines, so it may be running while the aircraft is on the ground. The APU may also provide conditioned air for the aircraft at the gate. The APU can be started using batteries or from the generators on the main engines. Some APUs have separate started batteries from the aircraft other batteries, depending on aircraft needs.
The APU is available for emergency needs. If the aircraft engines are unable to provide air conditioning and pressurization, the APU may be used for that. If the main generators have failed, the APU may provide electricity for the aircraft.
Various scenarios are possible. A maximum performance takeoff will require bleed air from the main engines to be cut off, and the APU can provide pressurization in that case. MELs may allow an aircraft to fly with a single operating generator if the APU is available. Sometimes both generators will fail, and the APU can take the load. For ETOP operations, the APU may need to be running for a portion of the flight (depending on operating limitations).
APUs fail, sometimes. The APU uses fuel, and people are very conscious of fuel consumption, so they are not used all the time. An APU may sit idle for days on certain aircraft, and when they are needed, they don't work, won't start, batteries are dead. The need is still there, so most aircraft have another backup device for emergency electricity generation, called the Ram Air Turbine (RAT).
The RAT is capable of powering enough of the aircraft to get it on the ground. This is a last final device for when things are going bad. The RAT is a propeller attached to a generator that will drop down into the slipstream air close to the fuselage. The electricity will be used to power the pilots PFD and whatever else the pilots need (IE fly by wire system).
The twin aircraft will have two generators, one on each engine. The APU will be there if one or more of those generators fail, and some aircraft have a RAT if all the other generators fail. There is proably no good reason for a pilot not to be able to land a plane if all the electrical systems are out. Will that help you fly more comfortably?