The schoolmaster is the teacher of the school where most of the kids in St. Petersburg attend. Many of the kids find him to be very cruel.
He actually wanted to be a doctor, but because he grew up at almost poverty level, he was unable to do so.
This is revealed, because Becky is looking through his medical book and accidentally rips a page. Tom takes the blame and on that day, he is whipped for spilling ink on his spelling book as well.
The schoolmaster apparently becomes very tense and strict before the examinations, so the students make a plan. On the day of the examination, they get the sign-builder's son to use a cat to take off the schoolmaster's wig, which embarrassed him, although he did not initially realize this.