The main sources of neutrons for the s-process (can be found in under a minute on wikipedia and in this case is quite reliable) are carbon 13 and neon 22. These undergo reactions with alpha particles yielding oxygen and magnesium respectively plus a free neutron. The first reaction takes place in low-mass (1-8$M_{\odot}$) stars whilst they are in the AGB phase (hydrogen and helium shell burning); the second takes place in higher mass stars at the end of carbon burning.

For the r-process, HDE226868's explanation is the consensus explanation.

The main sources of neutrons for the s-process (can be found on wikipedia and in this case it is quite reliable) are carbon 13 and neon 22. These undergo reactions with alpha particles yielding oxygen and magnesium respectively plus a free neutron. The first reaction takes place in low-mass (1-8$M_{\odot}$) stars whilst they are in the AGB phase (hydrogen and helium shell burning); the second takes place in higher mass stars at the end of carbon burning.

For the r-process, HDE226868's explanation is the consensus explanation.