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A 'quantum vibration' is a vibration of a chemical bond in a molecule that must be treated quantum mechanically. The low-lying vibration energy states can be described as states of the quantum harmonic oscillator, and at higher vibrational states, near the bond disassociation limit, as Morse oscillators.
A molecule can vibrate in many ways and each of them we can call a ''vibrational mode''. The vibrations can be seen with IR spectroscopy for example.
As a help to calculate the number of vibrational modes, it's convenient to determine the number of degrees of freedom available to vibration. As a generalization any molecule consisting of N atoms will have 3N ''freedoms'' for translational motion. 3 degrees are translational freedoms. There are also 3 degrees of rotational freedom for non-linear molecules and 2 degrees of rotational freedom for linear molecules. This leaves 3N-5 degrees of vibrational freedom for linear molecules and 3N-6 degrees of vibrational freedom for non-linear molecules.
As an example H₂O, a non-linear molecule, will have 3
★ 3-6 = 3 degrees of vibrational freedom, or modes.
Another example is the linear carbon-dioxide molecule, CO₂, which has 3
★ 3-5 = 4 degrees of vibrational freedom.