As has been told, the hadrons are made up out of quarks. But there is something peculiar about this. When one looks at the constituents of, say, an atom - the nuclei and the electrons - then one can observe all of these also as individual particles. However, this is not applying to quarks. It has not been possible to isolate a quark experimentally. The only thing which can be seen are the hadrons.
When analyzing the structure of the hadrons in search for the reason, it turns out that one can assign to a quark a new charge, the so-called color charge. The name is just fancy and the charge has nothing to do with color. This color charge comes in six types. There are three 'positive' charges, called red, green, and blue (some people occasionally exchange one of the names for yellow), thus carrying the metaphor further. They are like the positive electric charge. Then there are three 'negative' charges, anti-red, anti-green, and anti-blue. They are like negative electric charge. As for electric charges, a negative and a positive color (say anti-green and green) neutralize each other. The amazing difference compared to electric charges is that also three different colors of either type neutralize each other. For example, a red, a green, and a blue quark together are neutral with respect to the color charge.
It is then found that all hadrons are always color-neutral. The mesons are made from one quark with color and one with anti-color, and the baryons are made from three quarks, each carrying one of the colors. In fact, the ones we observe around us are all made of three quarks carrying color. Those which carry anti-color are actually anti-matter, which will be discussed later.
Now, this gives an idea why quarks and hadrons are different. But it does not explain why quarks are not observed. This is now du to the force acting between two color charges. In contrast to all other forces, this force is not getting weaker with distance, but stronger instead. And it gets so quickly stronger that it is not possible to tear a hadron apart into quarks. At least that is what it looks like at the surface. The truth is somewhat more subtle, and not fully understood, and part of my research. Therefore, I will come back to this question many times in the future.
Irrespective of this, the force between colored objects is mediated by gluons. In contrast to photons gluons carry themselves color charges, though they are of a different type than those of quarks, and there are eight different ones, not usually given a name. As a consequence, the enormous strength of the force also binds gluons, and they cannot be observed as freely roaming particles either. In fact, at least in principle it is possible that gluons alone form bound states, much like hadrons. These are called glueballs, but are up to now only hypothetical constructs which have not been observed in nature, though some observations may hint at them. It is an ongoing experimental endeavor to find them.
The justified question is, if the force is so strong, why do we know about quarks and gluons? And why can it still bind the nucleons to nuclei if the nucleons are color-neutral?
Well, the force is not strong at all distances. Indeed, it grows quickly with distance, but on the other hand it diminishes as quickly with shorter and shorter distances. To the best of our knowledge it even ceases completely if one would be able to reduce the distance to zero. That is called asymptotic freedom. Therefore, if one can send a probe close to a quark, then one can identify its existence. For this purpose it helps very much that a quark is not only carrying color charge, but also electric charge. Therefore, it can be registered more easily by hitting it with a photon or an electron. That is somewhat indirect, but that is one of the main sources of experimental information on the quarks. The gluons are even more complicated, since they only carry color charge. Therefore, our evidence for them is rather indirect.
This is then also how nucleons feel each other by the strong force. When they come close to each other, they start to see each others quarks, which then can interact by the strong force. This is a comparatively weak effect, since it is, vastly simplifying spoken, just a bit of penetration what makes them feel each other. Nonetheless, this remainder of the force is so much stronger than the electric force that it makes the nuclei about 100000-times smaller than an atom. This should give an idea of how very strong this force must be that even such a small glimpse of it has such far-reaching consequences.
It is this strong force to which I will return repeatedly, as it is and has been for a long time my major focus of research. The one reason is that our understanding of this force is not very good is because many approaches just have to give up when faced with such an enormous strong force. Only at very short distances we have reliable control over it. Hence, the theory of this force, which is called quantum chromo dynamics (for the Greek word chromos for color and short QCD), is very hard to tackle. There is a simpler version of it, which only deals with the gluons and neglects the quarks (and is therefore not a picture of nature). It is called Yang-Mills theory. Because it contains already many essential features of QCD, it often, and also for me, serves as a prototype theory for the strong force.