I was reading on hyperphysics about electric charge. It says that a proton has 1,836 times the mass of a electron, but they hold the same amount of charge. It made me think about why is it that one charge is negative and one charge is positive? What determines if a particle is negative or positively charged? is it a certain property within the particle or is it something else?

Negative and positive are arbitrary. The important point is that protons and electrons have opposite charge.

It's a property some particles have. We observe certain behavior for particles with these properties.

Another question related to this: why is charge quantised in units of the charge of the electron?

 

The only reason I an aware of that sounds plausible is the Dirac quantisation condition. This requires the existence of magnetic monopoles, which have not been detected in nature.

Sorry if this is a bit naive - but how can you reconcile charge quanitised in units of the charge of the electron with the +2/3 - 1/3 charges of the quarks?

The charge of quarks comes in multiples on 1/3 e. So we still have a quantisation of charge.

 

We also have quasi-particles that do not obey this quantisation. But these are not true elementary particles, so no violation here.