At the schools that fall under the University of California system, all professors must have at least a doctorate in something. There may be a few programs that don't abide by this since there aren't any doctorate degrees for that subject area (art?), but for the most part having a doctorate degree is required.
To be a professor in the University of California, you need to also be: (1) consistently creative (e.g., come up with new ideas, and publish!), and (2) be at least a national if not international expert in your field. Obviously, if you are not a tenure track professor, these criteria are more relaxed.
In terms of the medical side of things, we have MDs, PhDs, MD/PhDs, JDs, MD/JDs, DVMs, and DVM/PhDs teaching us. This includes the med school, grad school, and even undergrads. Some of the med school professors have started their own courses to teach undergrads, such as our gross anatomy course which is team taught by an MD/PhD and a PhD. While this one immunology course is team taught by several PhDs, MDs, DVMs, and MD/PhDs. At the hospital we have clinical professors, who are at least an MD....which ultimately translates to being an attending.
Finally, on some rare occassions, when a PhD student or even rarer a masters student leads an actual course, they may earn the title of lecturer, or student lecturer. Something like that. But thats pretty much the teaching things that I am aware of here at the UCs.
To conclude though, at the university level, you will most likely need a doctorate in something. But more importantly you better be good with what you do, and well known in what you are doing to get into a competative, well funded institution. My PI said that when he applied to his present position...20 something years ago, they essentially called up people and asked about you. How "famous" you are in your area = them feeling confident in you as a professor...since you are using their space to make them $$$ and prestiege.
