The Quantum level is a term used for activity that takes place on a sub-atomic scale. This quantum scale is so small, compared to normal physical bodies, that Earth scientists didn't discover it untill centuries of developing physics and technology.
As a measure for the difference between normal and quantum scale, the Planck constant is sometimes quoted.
In 2370, Commander Data attempted to explain to Lieutenant Worf the theory of Quantum levels, when he discovered that Worf had travelled to various different Quantum realities. (TNG: "Parallels")