Quantum mechanics (QM) is a modern physics theory that models the mechanics (physics of forces and actions) at the scale of atoms and subatomic particles. Its mechanical principles have pronounced differences from the mechanics of larger objects viewed in everyday life as modeled by Isaac Newton's theories (classical mechanics aka Newtonian mechanics). In particular, QM models physical actions naturally thought of as inevitable as merely having a certain probability.
Whereas in classical mechanics, certain quantities may have any value, zero or greater, QM in some cases imposes minimal possible values (quanta), and it was developed precisely for 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 mechanics of particles in some cases can be modeled by considering them to be not particles, but waves.
Quantum mechanics' strangeness has been described as less well-known, yet stranger than the apparent paradoxes of relativity. An example is quantum mechanical phenomena's dependence on probability, e.g., quantum tunneling. 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 might be 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 "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.
Quantum fluctuations are the result of the probability aspect of QM: there are activities with a probability of happening at any time, thus there are certain things that happen at random.