I found a device which has a ferrite core and 3 coils on it. It looks like some type of transformer. What would be its use? I found it in a compact fluorescent lamp.
You are looking at the transformer of a very elegant and basic resonant converter (a.k.a. Royer converter).
A very basic explanation is this: There are two windings with an equal number of turns that are used to drive two transistors, and there is one winding that connects to the lamp. During each switching cycle, at a certain current, determined by the core material and the number of turns of the winding that goes to the lamp, the transformer will saturate and thereby define the power that is transferred to the lamp. Each time a saturation event occurs, one transistor will block and the other transistor will start to conduct, which keeps the circuit oscillating.
The Royer converter was invented long before switch-mode power supplies were the norm for power conversion circuits and is still very popular. Considering its age, it is amazing that efficiencies of > 85 % can be achieved easily.
A good application note with many details can be found here: http://www.nxp.com/documents/application_note/AN00048.pdf.
There is also an article on Wikipedia: http://en.wikipedia.org/wiki/CCFL_inverter
Another good App'note: http://www.diodes.com/zetex/_pdfs/3.0/appnotes/apps/an14.pdf
Here's a good manual about fixing compact flourescent lamps (CFLs). It includes many schematics: http://www.en-genius.net/includes/files/col_081307.pdf
The original reference to the circuit is here: Bright, Pittman and Royer, “Transistors As On-Off Switches in Saturable Core Circuits,” Electrical Manufacturing, December 1954.
Transformers can have any number of coils. Usually only one of them is used to put power into the core, and is called the primary, while the others take power out and are called secondaries.
You can have as many secondaries as you need. One thing you can do with two secondaries is to make isolated supply sections. Another purpose is not so much to pull power out, but feedback signals, which can be used to drive an oscillator.
Of course I went and said 'usually' only one of them is used to put power in. There are cases where two coils can be used to put power in. So called Push-Pull amplifier sections do this, as do some DC to AC inverters.