Understand the Joule Thief circuit
Here is the circuit being discussed:



To understand how this works, consider each phase of operation over a whole cycle.

Let's start with everything off and power newly applied.  the transistor is off, so no collector current.  However, there is a current path thru the resistor, the transformer secondary (the left coil), and the base of the transistor.  This turns on the transistor, at least somewhat.

The transistor turning on brings the bottom side of the primary (the right coil of the transformer) to a lower voltage than the power.  At this point, the primary acts mostly like a inductor, so current builds up linearly over time.

This change in current flowing out of the dot end of the primary causes the secondary to try to make current flowing into its dot end.  This additional current thru the secondary also means more base current, since the two are in series.  More base current means the transistor turns on harder, which causes more current thru the primary, which causes more base current, which turns the transistor on harder, etc.

Due to this positive feedback, the transistor is turned on so hard that its collector voltage can't go any lower, usually about 200 mV.  That's called saturation.  At this point, the transistor C-E looks like a closed switch, except for the small saturation voltage.

With the fixed voltage of the supply minus the saturation voltage applied to the primary, it's current ideally builds up linearly.  However, it can't do that indefinitely.  One of two things happen: The current levels off due to the inherent equivalent series resistance of the primary winding, or the the transformer core magnetically saturates.  Either way, the secondary is no longer driven to create higher current.  Since a transformer works on the change of the magnetic field, the secondary stops providing additional base drive at all.

Now the positive feedback works to quickly turn off the transistor.  At first when the magnetic field levels off, the secondary no longer provides additional base drive, and the base current is just what the resistor allows to pass.  That is not enough to sustain the same collector current, so the transistor starts to turn off a bit.  This now reduces the magnetic field.  The secondary now actively opposes the base drive provided by the resistor.  The transistor allows less current, which causes more active negative base drive, which causes even less collector current, etc.

Due to this positive feedback, the transistor turns off hard quickly.  The primary now acts like a inductor.  Its current can't be turned off instantly.  The inductor makes whatever voltage it has to to maintain the same current instantaneously.  The voltage rises quickly, and when it gets to the LED forward voltage, the current flows thru the LED and the voltage stays roughly fixed.

The inductor now has reverse voltage across it, so the current thru it is decreased.  Eventually it gets to 0, the magnetic field stops changing, the secondary stops opposing the current thru the resistor, and we're back to about the state where power was first applied.