I've never heard of a 2-phase BLDC. I've only ever heard of 3-phase and higher. I'm pretty sure PC fans use 3-phase BLDCs since I have never heard of a 2-phase BLDC, and pretty much only the military and aerospace use more than 3-phases.
A single-phase to 3-phase converter is called a "3-phase inverter". THey run off of a single DC rail (your so-called single phase). Google or wiki "3-phase inverter", find out what you can, learn the stuff that is harder for us to explain that it is for you to learn yourself, and come back with specific questions that are more easily answered by us and harder to learn on your own.
DO you understand how a half-bridge works? Or an H-bridge (which is basically two half bridges)? An inverter is just a half-bridge, and a 3-phase inverter is just 3 half-bridges (one for each phase). A 3-phase inverter could also be considered as 1.5 H-bridges. THey are used for power supplies and motor drives which are pretty much the same thing, just different performance goals.
Common components in them are:
-flyback diodes
-gate drivers
-at least 6 power switches (often MOSFETs or IGBTs, though thyristors are also used sometimes for very very very high power applications).