The s-process, short for "slow process", is one of two main ways that elements heavier than iron are formed inside of stars. The other process is the r-process, short for "rapid process". Both processes are endothermic and rare, but they take place in very different situations.
Up to the elements of Iron and Nickel, stellar nucleosynthesis is a process of fusion of whole nuclei, and it releases energy. Since iron or nickel is the most stable nucleus, further fusion is energetically unfavorable. So heavier elements are formed by neutron bombardment of iron and nickel. This requires having a source of neutrons, which is another story in itself. And this happens slowly, meaning that the nucleus (usually) undergoes some type of radioactive decay between absorbing neutrons.
An atomic nucleus, for various reasons, has to maintain a balance of protons and neutrons. If it has too many protons, it will emit a positron, transforming a proton into a neutron. Too many neutrons, it emits an electron, turning the neutron into a proton. In the s-process, when the nucleus absorbs a neutron, it will decay (over a time period of anywhere from seconds to hundreds of thousands of years) into an element with a higher atomic number. It then, after another few thousands or millions of years, is impacted by another neutron. It can take a long time for an element to slowly gain nucleons and atomic number this way.
Even this slow process requires a very large, energetic star. The s-process is most common in stars that already have a high metallicity, and that have high temperatures: either blue giants or red giants. Both of these types of stars are rare. But they are still much more common (and much longer lived), than supernovae, the site of the r-process. Although I have tried to find numbers on it, there does not seem to be good data on which source, r-process or s-process, is more dominant in the formation of various heavy elements. Given the rarity of such stars, and the difficulty of modelling the inside of stars, all of the details of the s-process are something that our science can't currently tell us, even though the theoretical process is fairly clear.