The quark model is a framework that describes hadrons as being composite particles made of even smaller, elementary particles known as quarks. It is an evolution of the eightfold way that extends in geometric and algebraic results into a more extensive, more complete scheme.
At heart, the quark model like its predecessor is an application of group theory to particle physics. It recognizes that the eightfold way developed a strongly algebraic formulation using group-theoretical concepts we already know of and expands upon it. It posits the existence of at least three particles that were at the time undetected and were possibly undetectable by law of nature, and uses these as the foundation to build different combinations of these: the special unitary group. Combinations can be pairs (mesons) or triples (baryons).
The quark model is a very successful and explains almost all of our findings in a neat, concise framework. It does have some flaws however, as it still can't predict why some exotic particles are precisely the way they are, but when it comes to ordinary matter and the phenomena we find in the ample majority of nature, it is a resounding success.