Hardtails
No rear shocks. Generally lighter, simpler, and easier to maintain. Great for local riding.
Dual Suspension
Squishy & Grippy. Confidence-inspiring bikes engineered to ride tough terrain with the best mechanical advantages.
Urban MTBs
Rigid. No front or rear suspension. Maximum simplicity. Shreddy.  Ride it to work. Pedal it to the trail. Shred some laps.  Cruise home. Sleep. Repeat.
Bikepacking Rigs
"Choose your own adventure" bikes.  Ya'll wanna ride Sheltowee next month? 
Freestyle MTB
Bikes for jumping
Fatbikes
Bikes with tires 3" or more