What's the difference between "teacher" and "professor"?

Is one more formal then the other?

Teach is including a more general concept.

According to Longman Dictionary of Contemporary English

Teacher:

someone whose job is to teach, especially in a school

Professor:

American English: a teacher at a university or college

British English: a teacher of the highest rank in a university department

To see academic degrees visit here.

Also you may find it interesting to know:

• teacher someone who teaches as their job, especially in a school : a high school teacher

• principal ( also headteacher British English ) the teacher who is in charge of a school or college : The teacher sent him to the principal’s office.

• tutor someone who gives private lessons to one student or a small group of students. In Britain, a tutor is also a teacher in a university : They hired a tutor to help him with his English. | Your tutor will help you find a subject for your essay. lecturer someone who teaches in a university or college : University lecturers aren’t very well paid.

• professor a teacher in a college or university. In Britain, a professor is a high-ranking university teacher, especially one who is head of a department : She was professor of linguistics at Cambridge University.

• instructor someone who teaches a sport or a practical skill such as swimming or driving : He works as a ski instructor in the winter. | a driving instructor

• coach someone who helps a person or team improve in a sport : a professional tennis coach

• educator especially American English formal someone whose job involves teaching people, or someone who is an expert on education : Most educators agree that class sizes are still too big.

• trainer someone who teaches people particular skills, especially the skills they need to do a job : a teacher trainer | Many companies pay outside trainers to teach management skills to their staff.

• governess a woman who lived with a family and taught their children in past times : As a governess, Charlotte Brontë received twenty pounds a year.

Professor is more specific of teacher, as it is used when referring to a teacher of the highest rank in a college or university.

Informally, professor is used to refer to any instructor, especially the specialized ones.

_{[Reference: the New Oxford American Dictionary.]}