Oscillator for High Voltage Transformer: RLC or Blocking Oscillator
I'd call this a flyback.   This site: http://members.shaw.ca/novotill/PhotoFlashes1v5/index.htm  has a similar schematic.  For more authoritative sources, Google isn't cooperating quickly enough at the moment.  

The "blocking" oscillators I find in an equally inadequate search all show a DC blocking capacitor between the inductor and transistor's base, and the transistor biased on directly by a resistor.  But the basic principle is similar. 

For the circuit shown, start off by ignoring lead 5 of the transformer its associated parts.  Look at the inductor between 3 and 4.  Since it connects the supply voltage to the base, current will flow. Not immediately, since inductors initially oppose currents, but it will very soon flow.  That turns the transistor on.  That lets current flow through the 1-2 part of the inductor.  Again, not immediately, but flow it will after a short delay.  

Now the fun part: Transformers with current flowing through one side will pull a parallel-direction current through the other side.  There seems to be a small error in the schematic you show.  Typically the collector goes to one end of one coil, and the base to the opposite end of the other coil.  Sometimes dots are placed at the ends of the inductors to make clear what is meant by "same end". I'll assume a corrected schematic in the rest of this explanation.  (Check the link given at the start of this answer.)

The current flowing from 2 to 1 causes a current to flow from 4 to 3, opposed to the initial current that started the transistor.  Assuming a well-chosen turns ratio, this current will shut off the transistor.  That will stop the current in the collector coil.   Now we're back to the starting situation. Repeat forever.

So we have an oscillator.  What about the rest of the schematic with the diode and capacitor?   We take into account the part of the inductor on the right, with lead 5.  There are many more turns between 5 and 3 than between 1 and 2 - this is a voltage booster.   The varying collector current in 2-1 induces a same-direction current from 5 to 3. Or, think of it making a large negative voltage at pin 5 of the transformer.  This charges up the capacitor with a large negative voltage.  It's a simple rectifier/filter circuit to generate DC from an AC or pulse source.

BTW, this is not any kind of RLC oscillator.  The resistor is just to limit the amount of base current so the transistor doesn't fry.  There are no capacitors in the oscillator itself - it is only in the rectifier circuit making use of the oscillator's output.   What sets the frequency is mostly the design of the transformer and characteristics of the transistor, and also the supply voltage, and the resistor (though it's not in any sense a timing resistor).

This site http://mysite.du.edu/~etuttle/electron/elect37.htm  does a fine job explaining blocking oscillators. Like I said, similar but not exactly the same as flybacks. Transistor and tube versions are described.  Note the use of dots to indicated parallel-arranged transformer coils, and how the B and C of the BJT are wired to the transformer.

In contrast, Wikipedia's article on Flyback Converters shows (as of Nov 2013) diagrams using dots on opposite ends, to indicate how the wiring should go, but keeping the drawn schematic a bit neater. http://en.wikipedia.org/wiki/Flyback_converter   Another example of dots on opposite ends is at http://www.interfacebus.com/transistor-blocking-oscillator-circuit.html Also, the link given at the start of this answer, as well as the Wikipedia article on Joule Thief http://en.wikipedia.org/wiki/Joule_thief