Starting both engines on a Boeing 777200 fitted with General Electric GE9085B engines.

Quick Explanation
The engines are started using compressed air. At the engine there is a Turbine Starter. That's in basic terms a fan on a shaft. The air blows the starter fan around which, through a gearbox and shafts, makes the engine blades turn. When the starter has sped up the engine to a certain point, fuel is added and ignited and the engine starts.

More in depth explanation
Where does the air come from?
The compressed air comes from the Auxillary Power Unit, aka the APU. The APU is a small gas turbine engine that is located in the tail of the aircraft. Instead of producing thrust, the APU drives an Electrical Generator and an air compressor called a Load Compressor. The compressed air travels along a tube, or duct, to both engines and the air conditioning packs. The APU is quite small and is started using an electric motor powered by it's own battery. The generator on the APU is suffecient to supply all the main aircraft systems with electrical power. The compressed air can be also used to operate the air conditioning packs.

How does the starter work?
At the engines there is the Turbine Starter. It is essentially a turbine on a shaft. The compressed air passing over the turbine makes it turn. It is connected to the Engine Accessory Gearbox.
The starter has it's own reduction gearbox, meaning that the starter turbine spins many times faster than the starter output shaft, which increases output shaft torque. This allows the relatively small turbine of the starter to turn the large engine. An example of this would be the gearbox in a car, in first gear the car accelerates faster than in top gear.

Then what?
The Accessory Gearbox is connected to the main engine through a shaft. It has all the "accessories" on it, things like a Generator, a Hydraulic Pump, and a High Pressure Fuel Pump. By turning the gearbox with the starter, some of the internal engine blades start to rotate.

What do you mean, some of the engine blades?
The GE90 is a TWIN SPOOL gas turbine engine. This means that there is 2 shafts in the engine that the compressor and turbine blades are mounted on.

First we need to understand the difference in blades.
1. Compressor blades are in FRONT of the combustion chamber (where we burn the fuel). They squeeze the air molecules together, increasing both air pressure and temperature, adding energy to the air, before it enters the combustion chamber. .
2. Turbine blades are BEHIND the combustion chamber. The hot expanded air from the combustion chamber with now even more energy, does "Work" on the turbine causing it to rotate.

Compessor blades and turbine blades are mounted on the same shaft. That is to say the Turbines directly drive the Compressors.

So, back to the 2 shafts. There's the Low Pressure section and the High Pressure Section. The LP section consists of the Fan, the big one at the front, several sets of compressor blades, and some turbine blades. The HP section is several sets of compressor blades and some turbine blades. They are on different shafts so that each section can rotate at different speeds to be more effiecient. The LP shaft is longer than the HP shaft, and is also thinner. The LP shaft runs inside the hollow HP shaft.

Front to back
1. The Fan
2. LP Compressors
3. HP Compressors
4. Combustion Chamber
5. HP Turbine
6. LP Turbine
7. Exhaust

The Accessory Gearbox is connected to the HP shaft by the Gearbox Drive Shaft. When starting, the starter drives the gearbox, the gearbox turns the HP shaft and its associated compressors and turbines. The HP compressors start drawing air through the engine. The moving air makes the LP section start to rotate. When the starter has accelerated, and enough air is moving through the engine, fuel is added and ignited. The starter continues to aid the engine spin up. As the combustion process takes hold, and more hot gas is produced, there's a point when the engine becomes self sustaining. The turbines take over from the starter, driving the compressors to suck in even more air. The starter disengages, and the engine speeds up a little bit more to idle.