“Teacher” is the general word. “Professor” is a job title given by a university to the faculty. You can have the title of (among others) a teaching assistant, lecturer, assistant professor, associate professor, or full professor (from lower to higher rank). Only those with “professor” in the title are professors.

In the United States, there are no rules, only customs.

In most states, in secondary schools (pre-university), teachers are required to have licenses. The term commonly used for people who teach in a secondary school is "teacher."

"Teacher" is not used as an honorific. You do not say "Teacher Wilson." If you wish to be respectful, you use the ordinary honorifics: Mr., Mrs., Miss. and Ms.

In universities, the word "teacher" is avoided. The ranks are as stated by Phil. Everyone with the word "professor" in their title can be addressed as "Professor." For example, one would refer to "Professor Noam Chomsky."

Although it is not, strictly, a requirement, most professors hold a "doctorate" degree, often the Ph.D. Anybody with a doctoral-level degree can be addressed as "Doctor." In U.S. English, if someone is a professor and has a doctorate you use only the title of "professor." Unlike some European countries, we never say "Professor Doctor so-and-so."

I said "university" just to make it clear that I'm talking about "higher education," university-<em>level</em> education. Confusingly, to people from other countries, we call the first four years of higher education "college." The faculty of a "college" are usually called "professors." A college may be an institution that only offers four years, such as William College. A "university" contains within it many schools; perhaps a law school, a medical school, a music school, and various "graduate schools." The first four years at a university are provided by a "college" that is one of the schools within the "university."