I certainly don't know all the physiology involved, but there's a combination of hormones, enzymes, and various other stimuli. The cervix ripens, the baby moves down, bag of waters may break, dilation, etc. But then there are mental/emotional influences as well, and I've heard a lot of mind-over-matter type stories.
It is interesting that the timing of labor/birth is so different with different people. Babies arrive early, late, and on-time. My mom claims I was born a month late. Yikes! Guess it was nice and cozy in there and I didn't want to come out.