Conquer Cycling Trainers are compatible with 26", 27" and 700c wheeled bikes. Also, the rear wheel of your bike must have a quick release skewer. The trainer has its own skewer that replaces the one on your bike (making your bike, and many other bike models, able to work with it).
Regarding the trainer being compatible with a mountain bike, typically the thicker tread associated with mountain bike tires may prevent the bike from sitting in the unit properly. If this is the case, thinner tires may be used and that should resolve any issues, as long as the bike fits the (above) requirements needed to use the trainer.
Regarding "fat tire" bikes, the resistance unit on Conquer trainers can accommodate tires that are less than 2.75 inches wide (the silver part of the resistance unit where the rear tire rests is only 2.75" wide).