Very simply, it's the study of matter at the very very smallest (atomic) level. The reason for giving it its own category of study is because at the quantum level, things operate very counter-intuitively from what we're used to experiencing in classical mechanics at the macro-level.

The basis for the term quantum mechanics is the "quanta" or the idea of the very smallest unit of matter. For an example, matter is typically made up of collections of molecules, which are in turn made up of collections of bound atoms, which iare n turn made up of collections of electrons, protons, and neutrons, which are in turn made up of quarks and neutrinos, and as far as we know at the moment, those aren't made up of anything else.

When you start running experiments with single bits of matter or energy, such as a single electron, or a single atom or photon for example, you encounter some interesting behavior such as a dual behavior of matter as both discrete particles (including mass and effects of gravity) and as a wave (such as diffraction interference patterns). Quantum mechanics can explain this dual particle-wave nature, where classical mechanics cannot. For other behavior, such as the position of an electron in a stable nucleus orbit, quantum mechanics provides a prediction of probabilities, compared to classical physics, which would predict certainties.
Source(s):
http://en.wikipedia.org/wiki/Quantum_mechanics

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