Decades ago particle physicists were observing new subatomic particles in the laboratory faster than they could predict or explain them. In order to make some sense of this emerging pattern, scientists had to suppose that these particles were made of a smaller set of building blocks: quarks.
In the same way earlier that the numerous atoms on the periodic table of elements had pointed to the existence of more "element"ary subatomic particles, the production later of numerous subatomic particles was a clue that we had another sublayer of elementary particles to go.
On the following Web page, you will see symbols for many of those particles. By clicking on a symbol, you can observe a typical decay of that particle. If the particle is a hadron, you can see the quarks that are inside it. There are no quarks shown for some of the particles (leptons and bosons), because as far as we know, they are as elementary as quarks themselves.
If you want to see a list of particles along with the quarks that are inside of them, just go to the particle table. If you would like to know the definitions of some unfamiliar terms, try the glossary at Boston University.