Starting both GE90115B engines on a Boeing 777300ER. These are currently the largest and most powerful gas turbines fitted to an airliner. (2015)

Type: axial flow, twinshaft, bypass turbofan engine
Length: 287 inches (7,300 mm)
Diameter: overall: 135 inches (3,400 mm); fan: 128 inches (3,300 mm)
Dry weight: 18,260 pounds (8,280 kg)
Thrust: rated at 115,540 lbf (514 kN).
Equivalent to ~145,000hp (GE Website)
Cost: $2224 million (2009 USD)

Quick Explanation
The engines are started using compressed air. On the engine there is a Pneumatic Starter Motor. 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 sufficient 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 Pneumatic Starter. The starter is a unit that is about a foot in diameter and about a foot long. It is connected to the Engine Accessory Gearbox. Inside it is essentially a turbine on a shaft. The compressed air passing over the turbine makes it turn.
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, the engine spins many times faster than the wheels.

Then what?
The Accessory Gearbox is connected to the main engine through a shaft. It has all the "accessories" on it, things like an Electrical 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. The compressors are said to "do Work on the air".
2. Turbine blades are BEHIND the combustion chamber. The hot expanded air from the combustion chamber with now even more energy, pushes on the turbines causing them to rotate. Now the "Air does Work on the turbines".

Compressor 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 efficient. 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 purpose of the HP turbine is to extract enough energy to turn the HP compressor. There's still lots of energy left over, the LP turbine extracts most of it to turn the Fan, and the LP compressor. The Fan produces around 80% of the engines thrust.

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.