Quantum mechanics (as opposed to classical mechanics) is that part of physics where Planck's number $\hbar$ cannot be neglected. It covers all quantitative microscopic physics (below the level of molecules) but also certain aspects of macroscopic physics such as superconductivity.
Quantum mechanics is characterized by the fact that not all observable quantities (considered as part of an algebra) commute with each other. As a consequence, the collection of states (positive linear operators on the observable algebra) has a structure different from that familiar from classical mechanics: Even in pure states (states that cannot be written as mixture of other states), most observable quantities have an intrinsic uncertainty. Most familiar is the Heisenberg uncertainty relation that specifies a positive lower bound on the product of the uncertainty of position and momentum in nonrelativistic quantum mechanics.