This is a 1.2-volt single transistor flyback (Joule Thief) circuit that features a third coil. With it, flash duration and brightness is much enhanced, even with just a 10uF capacitor, as can be seen in the schematic.
The parts to the right of T1 form a simple Joule-thief (‘Blocking’ oscillator) circuit which boosts the 1.2v supply to 3.5v to operate the LED. C1 is used to extend the charge and discharge cycles to increase brightness and efficiency. Components R, C and D1 form a simple timing circuit which, with the values shown, is about 1 sec per flash. Capacitor C charges up through resistor R until its voltage is enough to bias D1 and turn Q1 on to light up the LED. The charge on C is slowly drained, until it is no longer able to keep Q1 on, at which point the LED turns off until C can be recharged by R again. More information on this circuit and others can be found at http://quantsuff.com.