Question: In American schools what do they mean by home room teacher? 

Answer: When the first school bell rings the students go to their home room.  This is a general room for the first of the morning where announcements are made and general administration stuff is completed like attendance. The teacher in charge of this class is the home room teacher.  When the first class bell rings, students will leave the homeroom to go to other classes such as history, math, music, etc. unless they have a class in the homeroom by where a teacher will come into the class to teach a particular subject.  This teacher may or may not be the homeroom teacher.
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