How are turbine engines started?
The means of starting most turbine engines is high pressure bleed air.  This bleed source typically comes from an APU but may also come from an already running engine (crossbleed start) or an external huffer cart.

In some airplanes the start is completely FADEC controlled and need only be initiated and monitored by the flight crew, and in other airplanes certain steps must be manually initiated.

The process below best describes a turbofan engine.  Read @Falk s answer for information on differences with turboprop engines.

Turbofan basics

Refer to the image below (Credit: K. Aainsqatsi, Wikipedia): 

The portions of the engine I will be referring to below are:


The N1 fan, N1 shaft and N1 turbine which are displayed in green and labelled "Fan/Low-pressure compressor", "Low pressure shaft" and "Low pressure turbine".  These units are connected and move as one piece.
The N2 compressor, N2 shaft and N2 turbine, displayed in purple and labelled "High pressure compressor", "High Pressure shaft" and "High pressure turbine".  These are connected and move as one, but independently of the N1 shaft.  Un-pictured is an accessory drive that is geared to the N2 shaft to drive engine accessories.
The hot section, pictured yellow and labelled "Combustion chamber".  This is home to a constantly burning fire in a jet engine. 


Engine Start

The process of the engine starting follows this basic formula


Through the opening of bleed air valves, bleed air is sent to an air turbine starter.  These devices typically use the high pressure bleed air to spin and engage a centrifugal clutch connected to the engines accessory drive.  This in turn causes the N2 shaft within the engine to spin.
With the N2 shaft spinning, the N2 compressor and the N2 turbines are spinning.  This begins to force air through the engine from front to back.  
With the accessory shaft and N2 shafts spinning, accessories should start working and this can be verified by oil pressure indications on the EICAS.
With increased N2 rotation, ignition will be turned on.  These igniters are located in the hot section of the engine and produce small sparks.  There should be an indication on the EICAS that the ignition is active.
With further increase in N2 rotation, fuel flow will be introduced.  This will be verified on the EICAS.  Once fuel flow is noted, it is important that the next stop happen fairly soon.
Light off!  The fuel is lit by the ignition and now the fire burning in the hot section supplied by air from the compressor is producing thrust across the N2 and N1 turbines.
As the engine is producing thrust across on the N1 turbine, the N1 shaft is spinning the N1 fan and the EICAS will note this increase in N1 rotation.  N1 and N2 rotation speeds increase.
Above a N2 threshold, bleed air valves supplying the the air turbine starter will close and the starter disengage. The igniters will turn off at some N2 threshold.  
The engine will settle into a stable idle thrust setting.


What can go wrong?


Hung start


N2 fails to spin up sufficiently for fuel flow to be introduced.  
Each start has a time limit and the start will be aborted.

No ignitors


Abort the start and switch igniters.  
Give MX a call.

Failure to ignite


Fuel flow is introduced but light off has not occurred in a proper timeframe
Fuel is building up in the engine and if light off does occur, it can be damaging.
The start is aborted and dry motoring is performed to clear the engine.

ITT exceedence


There is usually a limitation in the inter-turbine temperature during engine start.
If this is exceeded the start must be aborted
This may require coordination with MX before another start is attempted.

The air turbine starter fails to disengage


This must be corrected before departing
In some engines, MX can correct this with the engine running.

The bleed air shutoff valve supplying the air turbine starter fails to close


This may require an engine shutdown to remedy
However, in some airplanes, this valve can be manually shut by MX with the engine running
Coordination with MX required.

Runaway N1 or N2 


If the FADEC isn't managing fuel flow properly, the engine may not settle but continue to spin up.
Abort the start.
Coordinate with MX before attempting a restart.



What about starting the engine in the air?

If the engine suffers a flameout, an airborne restart may be attempted.  These starts typically happen one of a few ways:


Crossbleed start
APU start
Windmilling start


The APU start is essentially the same process as above.  The crossbleed start, which can also be done on the ground, merely substitutes a running engine at a high power setting to provide the bleed air for starting and is otherwise the same as above.

The interesting start is the windmilling start.  The necessity for this means something bad has happened.  To need a windmilling start, this means that there are no bleed air sources to supply the air turbine starter.  This can mean that all engines are out and the APU is unavailable (BAD!), or merely that bleed valves to a shutdown engine have failed closed and are unable to be opened.

For the EMB-145 that I am familiar with, a windmilling start required descending at an airspeed between 260 KIAS and 320 KIAS and could not be attempted above FL250.  In short, you hope that the mass flow through the engine is enough to spin the N2 compressor as the ATS would.  With an N2 indication within the engines airstart envelope, you introduce spark and fuel and hope that the engine lights.  In the worst case, if you too slow and unable to provide enough airflow prior to light off, the engine can quickly overtemp and be damaged.  For this reason it is especially important to abort this kind of start as soon as an abnormality is detected.