반 is the group of students in a room (they usually stay in one class the whole day). Example: 3-7. This would be 3학년 7반. Grade 3, room 7 (or literally class 7).
Let's take look at the hanja...學級, 학 is easy, used for something related to study. 급 means 등급 as in level/ranking. Looking at example sentences (I've maybe heard this word used once or twice in 2.5 years here), it looks like it could be used as 수업 or 빈 or 학년 almost like it's a general term. I've copied a few below.
그는 항상 학급에서 1위이다. - He's always at the top of his class. 이 학급에서 누가 가장 나이가 많지? - Who's the oldest in the class? 그는 성별로 학급을 나누었습니다. - He split up the class by gender. 영어의 초보자를 위하여 학급을 만들다. - I'm making a class for beginners English.