Hi Jessica,
this is quite a hot topic of research, so a lot of the knowledge about what capsaicin does is incomplete. One way it is thought to stimulate apoptosis is through direct induction of the p53 tumor suppressor gene (published in the last couple of months). However, capsaicin also affects a variety of other pathways in cancer cells, and that is certainly not the end of the story.
Capsaicin is a vanilloid, which binds to VR1 (a vanilloid receptor), and stimulates nerve cells - VR1 is an ion channel which allows ions to flow into the cell. VR1 is also activated by heat, which is why capsaicin in peppers feels like a burning sensation on the tongue. So you might think it would increase the pain ! However, prolonged exposure to capsaicin appears to desensitize neurons to pain stimuli - possibly because it blocks other ion channels in the cell, or depletes the supply of neurotransmitters. On top of this, capsaicin seems to destroy a molecule called DSP, which itself damages cartilage in joints.
Quite a molecule to come out of chilli peppers !
I hope that will give you enough information to do some further research through a resource like Google scholar:
http://scholar.google.com