Quarks are fundamental matter particles that are constituents of neutrons and protons and other hadrons. There are six different types of quarks. Each quark type is called a flavor .
Quarks only exist inside hadrons because they are confined by the strong (or color charge) force fields. Therefore, we cannot measure their mass by isolating them. Furthermore, the mass of a hadron gets contributions from quark kinetic energy and from potential energy due to strong interactions. For hadrons made of the light quark types, the quark mass is a small contribution to the total hadron mass. For example, compare the mass of a proton (0.938 GeV/c2) to the sum of the masses of two up quarks and one down quark (total of 0.02 GeV/c2).

So the question is, what do we mean by the mass of a quark and how do we measure it. The quantity we call quark mass is actually related to the m in F = ma (force = mass x acceleration). This equation tells us how an object will behave when a force is applied. The equations of particle physics include, for example, calculations of what happens to a quark when struck by a high energy photon. The parameter we call quark mass controls its acceleration when a force is applied. It is fixed to give the best match between theory and experiment both for the ratio of masses of various hadrons and for the behavior of quarks in high energy experiments. However, neither of these methods can precisely determine quark masses.   

How Do We Know Quarks Are Real?

A question you might well ask! If we cannot separate them out, how do we know they are there? The answer is simply that all our calculations depend on their existence and give the right answers for the experiments.

For example, when we bounce electrons off of protons and neutrons, the pattern of scattering angles observed is characteristic of point-like spin-1/2 scatters. The relative rates for electron versus neutrino scattering is that predicted from the quark electric charges. The process of electron-positron annihilation to quark pairs gives similar characteristic predictions, all these are also confirmed experimentally. The accumulation of many such results, where experiments match predictions based on quarks, convinces us that quarks are real.