Matterwave basically has it correct; the main sources of elements in the Universe heavier than iron is from the r-process and the s-process. They are split more or less 50:50 between these two processes.
The r-process is still being researched, but observations provide fairly convincing evidence that massive stars are the source. Note that although the r-process is poorly understood, models exist that model high-entropy winds and/or neutrino winds that predict interesting r-process abundance signatures.
The s-process comes from a variety of sources, namely AGB stars and massive stars during helium burning.
For AGB stars, a specific a scenario can occur where a ^{13}C pocket is formed, which can undergo the reaction ^{13}C(\alpha,n)^{16}O during a thermal pulse. The neutrons are then absorbed by everything around, but mainly Fe-group nuclei since they are so abundant. The abundance signature created by this mechanism, averaged over all AGB stars of slightly different masses, gives rise to the 'main-component' of the Solar system abundances.
For massive stars, the reaction chain ^{14}N(\alpha,\gamma)^{18}O(\beta^+)^{18}F(\alpha,\gamma)^{22}Ne creates a fair amount of ^{22}Ne during helium burning. This ^{22}Ne then acts as a neutron source due to the ^{22}Ne(\alpha,n)^{25}Mg reaction. This requires hotter temperatures than the ^{13}C neutron source to activate, so it doesn't really activate until carbon shell burning. This process gives rise to the 'weak-component', which is effectively a correction to the main component.
Note that both of these processes depend on the metallicity of the star.
There are many other nucleosynthesis processes too, such as synthesis of ^{208}Pb in lower mass stars (s-process giving rise to the 'Strong-component'), the p-process (proton-captures), \nup-process, weak r-process (low/zero metallicity equivalent of the r-process)... the list goes on! Some of these are very recent developments.