The teachers who teaches in Government schools are called Government Teachers. In this case, the candidate have to apply for school service examination organized by state government. On qualifying that exam along with interview, the panels are released and rank wise the candidates are recruited as government teachers.

Similarly teachers who used to teach in Private schools are called Private teachers. Also some times, the teacher who gives private tuition are also called private teachers. Usually teachers in private schools are released by the notification released by the institution itself followed by an interview. Also for Government teachers B.Ed is compulsory but in case of Private institutions it may not be compulsory.

I hope this helps.