Although it may seem counter intuitive, the first quantum number we should discuss is spin. Sometimes associated with the top like rotation of a subatomic particle about its central axis, in truth spin is a method for sorting the fundamental particles. Like the characters in a video game may be divided into groups (the different types associated with each Pokémon, perhaps) the different particles of the Standard Model are divided into two fundamental families according to their spin: fermions and bosons.
Particles with half integer spin (1/2, 3/2, 5/2, …) are fermions. These guys are characterized by their tendency to be totally “standoffish”, which is to say they really don’t like to be near one another (more precisely, they cannot occupy the same quantum state). Fermions are the building blocks of all ordinary matter.
The set of fermions is further comprised of two particle subtypes: leptons and quarks. The difference between leptons and quarks is that the former constitutes a complete particle in its own right, while the latter can only exist in groups of two or more. The astute observer who noticed that the proton and neutron were missing from the Standard Model will be happy to hear that each are present, in spirit, as combinations of quarks – up, up, down, and down, down, up respectively.
The goal of the Standard Model is to understand the most basic building blocks of matter; therefore, particles like protons and neutrons, called hadrons, which can be decomposed into smaller constituents, are not included. In principle we can understand hadrons just fine if we can understand all the pieces that go into making them. Just beware not all hadrons are fermions! But, if a particle is not a fermion, what is it?
Particles with integer spin (0, 1, 2, …) are bosons. Bosons, in contrast with fermions, are on speaking terms with all other bosons and thus have no objection to occupying the same space (quantum state). Where fermions form the basis for the matter of the universe, bosons have a more intermediary role as the carriers of the fundamental forces of nature.