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Quantum mechanics (QM) is a 20th century physics theory that models the mechanics (physics of forces and actions) at the scale of atoms and subatomic particles (termed quantum scale). Its mechanical principles have pronounced differences from the mechanics of larger objects viewed in everyday life modeled by Isaac Newton's theories (classical mechanics aka Newtonian mechanics). In particular, QM models physical actions that are naturally thought of as inevitable as merely having a certain probability. QM's development began after experimental results regularly "made no sense", e.g., the same situation produced varying results that showed consistency only in their statistical frequency of the outcomes. Mathematics capable of calculating the observed frequencies was developed, and that math is termed quantum mechanics.
Whereas in classical mechanics, certain quantities may have any value, zero or greater, QM in some cases imposes minimal possible values (quanta), and was developed precisely to include this feature when certain observed phenomena could be explained by such a trait. The notion of quanta of light (photon) was developed to explain observations about the photoelectric effect, and quanta of angular momentum to explain why an electron's orbit around a nucleus has a minimal size that remains sufficiently stable so as not to (immediately) decay into a merger with the nucleus.
Light quanta act like particles, resulting in two apparently-conflicting models of light: as waves and as particles, and consolidating these aspects results in some of the strangeness of QM. Subsequent study of the mechanics of atomic and subatomic particles revealed an analogous particle/wave duality, i.e., the behavior of particles in some cases can be modeled by considering them to be not particles, but waves.
Quantum mechanics' strangeness has been described as not so widely known, yet stranger than the apparent paradoxes of relativity. An example is quantum tunneling: a particle occupying a position it didn't have the kinetic energy to reach. Quantum mechanics has also been described as far more influential than relativity to current daily life, given it provides the successful explanations of the workings of transistors (thus virtually all current electronics), though it can be argued that workable transistors could conceivably have been developed through trial and error.
Quantum mechanics has a traditional interpretation that is at odds with our everyday experience and our notions of logic, and debates as well as perhaps a whole branch of philosophy attempt to address the problem of that interpretation's meaning and consequences, as well as other possible interpretations. In any case, QM calculations do produce numbers that match observation whereas other methods fail despite years of expert efforts to come up with something different, and an attitude of QM users has been not to dwell on debates and philosophy because it works (i.e., "shut up and calculate").
The term quantum system refers to a bunch of things having a quantum-mechanical interaction, i.e., in a manner modeled by QM. Examples might be the workings of an atom, or the incident of an electron meeting a photon. QM predictions work best if the system has some separation from other particles, e.g., a nearby free electron can affect the calculations that model the behavior of an atom. A quantum system has a quantum state, essentially the values of its quantum numbers. (This "system" aspect is like classical mechanics, in which, for example, the orbit of one body around another can be modeled neatly if no other body is very close: otherwise interactions of three bodies must be considered, i.e., the relevant system includes the third body as well.)
Quantum mechanics was developed in two stages once it was realized that some physical quantities are subject to quantum limitations. The term quantum mechanics is generally used specifically for the second version, which was developed in the 1920s and includes the Heisenberg uncertainty principle. The term quantum mechanics is also used to distinguish it from subsequent, more-developed theories that explain additional phenomena: some of the latter are termed quantum field theories (QFTs), among them quantum electrodynamics (QED) incorporating electric fields and magnetic fields, and quantum chromodynamics (QCD) incorporating the strong force.