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The proton’s world is a totally quantum one, and so it is described entirely by just a handful of numbers, characterizing its spin (a technical term, not to be confused with the everyday English word; the proton’s spin is 1/2), electric charge (+1 e, or 1.602176487(40)×10-19 C), isospin (also 1/2), and parity (+1). These properties are derived directly from those of the proton parts, the three quarks; for example, the up quark has an electric charge of +2/3 e, and the down -1/3 e, which sum to +1 e. Another example, color charge: the proton has a color charge of zero, but each of its constituent three quarks has a non-zero color charge – one is ‘blue’, one ‘red’, and one ‘green’ – which ‘sum’ to zero (of course, color charge has nothing whatsoever to do with the colors you and I see with our eyes!).
Murray Gell-Mann and George Zweig independently came up with the idea that the proton’s parts are quarks, in 1964 (though it wasn’t until several years later that good evidence for the existence of such parts was obtained). Gell-Mann was later awarded the Nobel Prize of Physics for this, and other work on fundamental particles (Zweig has yet to receive a Nobel).
The quantum theory which describes the strong interaction (or strong nuclear force) is quantum chromodynamics, QCD for short (named in part after the ‘colors’ of quarks), and this explains why the proton has the mass it does. You see, the up quark’s mass is about 2.4 MeV (mega-electron volts; particle physicists measure mass in MeV/c2), and the down’s about 4.8 MeV. Gluons, like photons, are massless, so the proton should have a mass of about 9.6 MeV (= 2 x 2.4 + 4.8), right? But it is, in fact, 938 MeV! QCD accounts for this enormous difference by the energy of the QCD vacuum inside the proton; basically, the self-energy of ceaseless interactions of quarks and gluons.
Further reading: The Physics of RHIC (Brookhaven National Lab), How are the protons and neutrons held together in a nucleus?, and Are protons and neutrons fundamental? (the Particle Adventure) are three good places to go!
Some of the Universe Today articles relevant to proton parts are: Final Detector in Place at the Large Hadron Collider, Hidden Stores of Deuterium Discovered in the Milky Way, and New Study Finds Fundamental Force Hasn’t Changed Over Time.