A quantum number is a number used in quantum mechanics that represents an amount of some quality that is preserved through interactions between particles, e.g., if two particles combine, then their resulting particle has the sum of their individual quantum numbers representing that quality. Atoms and subatomic particles have quantum numbers of different types (i.e., for different such qualities) and through the interaction of particles, including splitting or combining, the sum of each type of quantum number remains the same. This is analogous to the physics concept of the conservation of energy: that total energy remains the same through an interaction, but quantum numbers deal with finer detail.
There are a number of "well known" quantum numbers associated with normal matter, electrons, protons, neutrons, and atoms, but those are for the quantum mechanical description of interactions of interest.
Examples of interactions in which quantum number conservation is of interest include emission of a photon by an atom whose electrons have a more-than-minimal state of excitation, the analogous absorption, photoionization, fission, fusion, and beta decay.
Examples of quantum numbers are spin and electric charge. Quantum numbers are generally specified so as to always be integral, though spin is defined such that the smallest unit is termed 1/2. (See "electron orbital" for more regarding quantum numbers associated with electron orbits.)