Does winding a choke inductor produce a transformer?
You might be able to estimate the turns ratio by putting a sine wave across the primary of the transformer, then measuring the unloaded AC voltage on the secondary. The ratio of voltage should be proportional to the turns ratio.

You could probably get a better answer by sweeping the frequency and taking the max value.

The circuit works by switching the MOSFET (which you've half-included) on and off to generate an AC voltage. This is a very basic form of flyback converter (although it's missing a diode, so the output will be AC).

You can read about flyback converters here: http://www.dos4ever.com/flyback/flyback.html







simulate this circuit &ndash; Schematic created using CircuitLab

Update:
I got curious, then confused, then determined.. I think your circuit may have a problem... Check out this simulation. I think it does what you're wanting. Leaving out the diode gave me some pretty gnarly looking AC output voltage. Also note the addition of C2 to 'catch' the current from the collapsing magnetic field in the transformer primary. In this simulation, Vsw peaks near 60V, which would not be terribly good unless you're using a suitably rated MOSFET.