What promise does Scout make in Chapter 2 of To Kill a Mockingbird?
Although it is not exactly a promise, I think that what you are talking about here is what Jem makes Scout understand at the beginning of the chapter.
Scout is just going to be starting to go to school with Jem -- this is her first year. Jem wants to make sure that Scout is not going to try to play with him at school. Presumably this is because he is older than she is and he would be embarrassed to be playing with his little sister.
Scout doesn't exactly promise not to do it, but it is clear that she understands she is not supposed to play with Jem at school.
I'm not so sure Scout made any promises in Chapter 2 of To Kill a Mockingbird. Jem made it clear to her that she was not to have anything to do with him while they were at school: "I was to stick with the first grade and he would stick with the fifth."
However, in Chapter 3, Atticus gets Scout to promise him that if she continues to go to school, they would continue to read together each night. It must be a secret kept from Miss Caroline, though, since Atticus "wouldn't want her after me."
Scout gets in trouble at school because she anounces she already knows how to read. Her teacher insists that she must stop because she needs to learn a specific method for reading at school. Scout is upset about being told to not read and enjoys sitting on Atticus's lap reading in the evening. Atticus makes a pact with her to keep her reading at home a secret from the teacher. Scout, however, must continue going to school.