That's a pretty broad question that is probably better covered elsewhere on the web however I'll try to give you some pointers.
An IGBT is really a "darlington" like device where the first stage is a MOSFET driving the base of a power bi-polar transistor.
Power bi-polar transisors and mosfets each have their own characteristics which differ and are more appropriate depending on the circumstance.
A bi-polar transistor will, when saturated, produce an almost fixed voltage across the collector-emitter junction under a wide range of currents. For high current applications this voltage can be a lot lower than across the drain-source resistance of a MOSFET, and as such, less power is dissipated, and lost, through the device.
However, bipolar transistors need to be driven with a current that is proportional to the load current. When driving large currents that base current becomes large enough to become problematic for whatever is driving it.
MOSFETS on the other hand are voltage driven. The driving current required is only proportional to the charge current required to drive the gate to the appropriate voltage. When on the device presents itself as a low resistance. The voltage across that resistance, and consequently the power dissipated and lost to the device, will vary much more significantly with load current.
IGBTs basically combine the better effects of both devices at the cost of switching time. By providing a MOSFET/Bi-Polar darlington you get the voltage driven low Vce characteristics you might need when switching large currents. The MOSFET is used to deliver the large base current to saturate the bi-polar transistor.
simulate this circuit – Schematic created using CircuitLab
However, that compromise comes at a cost of switching time. The effective gate capacitance of an IGBT is normally around an order of magnitude larger than that of a MOSFET on it's own.
Which is best to use for a particular application really depends on what and how you are trying to drive through the load.
Design Considerations: Start out with your required currents. If you can find a simple MOSFET that, when on at that current, produces close to the same voltage across it that a saturated bipolar-transistor will produce, or less, you do not need to use an IGBT. If your voltage drop is too high with a straight MOSFET design, then consider an IGBT but be very aware that the driving circuit needs to operate at a considerably slower frequency.
Ultimately: You need to do your due-diligence and look at part specifications to determine which available and affordable device wastes least power / dissipates least heat, and will work at the frequencies you plan on designing to. There are no shortcuts here.