Quantum mechanics is a theory in modern physics 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 physics theories (classical mechanics aka Newtonian mechanics). In particular, it 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, quantum mechanics 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 (photons) 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 as particles, resulting in two apparently-conflicting models of light: as waves and as particles, and consolidating these aspects resulted in some of the strangeness of quantum mechanics. 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.
The term quantum system refers to a bunch of things having a quantum-mechanical interaction, i.e., under study in the science of quantum mechanics. Examples might be an atom, or an electron meeting a photon. Quantum mechanics predictions work best if the system has some separation from other particles, e.g., a nearby free electron can affect the calculations you would carry out to model the behavior of an atom. A quantum system has a quantum state, essentially the values of its quantum numbers.
Quantum fluctuations are the results of the probability aspect of quantum mechanics: activities have a probability of happening at any time (a rare action might be a decay that has a long half-life, and a physical action that is an immediate consequence of previous activity merely has a probability so high as to be virtually inevitable), thus there are certain things happening at random.