A bit of own work using the "magic tools" like wikipedia or google would be really nice. Or some reading in lecture notes, research in the library and alike.
To cut things short (and finally give an answer):
The main alternatives for power semiconductors are MOSFET and IGBT. (There are others for 'special applications', but the "mass market" conentrates on these two.)
The conduction losses for MOSFET are P = ID2 * RDS
Conduction losses for IGBT: P = ID * VDS (VDS is about 2 V under normal conditions)
Taking into consideration the current limits (which evolve every 3 months or so), there is always a point where the losses in the IGBT are lower than those in the equivalent MOSFET.
Typically, MOSFETs with a rating beyond 600 V are hard to get while 1200 V for IGBTs are not unusual. So the arguments for IGBTs are voltage (where MOSFET are not available) and high currents where the losses in IGBTs are lower than in MOSFETs.
There are more factors to be considered (price, die size, switching frequency etc.), but the current trend is - only partly justifiable - vs. the IGBT.