What are the different classifications of sub-atomic particles?

Firstly, particles can be split up into to main categories: hadrons and leptons.

Hadrons interact with the strong force and are made of quarks. Leptons don't interact with the strong force and are fundamental particles (they aren't made of quarks). Examples of leptons are electrons, muons and neutrinos.

Hadrons can further be sub-divided again: Baryons and Mesons. 

Baryons are made up of three quarks, where as mesons are made up of a quark and anti-quark. Examples of baryons are protons (up up down) and mesons are pions (up antidown).