Her real name is Jean Louise Finch.
We first find this out in Chapter 2 when her brother Jem says that her name is Jean Louise Bullfinch. Obviously, he is joking about the last name. But we know that he is not joking about the first name and the middle name. We know this because Miss Caroline, the teacher, calls her by those names a few times later on in the same chapter.
As far as I know, we are not told why she is called Scout. Jem's nickname is obvious, but Scout's is not.