Atticus's main literal battle is for Tom's freedom during the trial. Because of Bob Ewell's negative reputation and because of the circumstances of the case, Atticus knows that Tom is innocent, but he also knows that it will be an uphill battle to convince an all-white, male jury (common for the time and place) to acquit a black man who's been accused by a white man of raping a white woman. Despite Atticus's strong defense, the circumstances and past precedent are too much to overcome. After Tom's guilty verdict is read, Jem asks Atticus:
"How could they do it, how could they?"
"I don't know, but they did it. They've done it before and they did it tonight, and they'll do it again and when they do it--seems that only children weep" (Lee 244).
Atticus's answer refers to a historical pattern, and it leads into his bigger, more figurative battle- the fight against a stubborn society. Many people say that To Kill a Mockingbird is a book about racism, which it is, but there's more to this statement. There are some racist characters (like Bob Ewell) and racist moments (like the near lynching of Tom) in the novel, but many of the characters (and arguably some who also sat on Tom's jury) are not guilty of the kind of racism which would produce violence or demonstrate clear hatred of another based on skin color, but rather it's a more subtle, institutional racism. Institutional racism shows itself in the form of white men who feel they have to take another white man's word over a black man's, even if the evidence presented goes against that conclusion. They do this because the racism has become a part of the law process, a part of the legal institution. I'd be inclined to argue that most of Tom's jury doesn't believe he's guilty, but they convict him anyway, because it's the way things have always been done.
Atticus's broader, figurative battle is against a town that's slow to change and a lot of townspeople who maybe are not overtly racist but are also not willing to take a stand against tradition (and thereby risk their reputations). As Scout told us in the very first chapter:
"Maycomb was as an old town, but it was a tired old town when I first knew it... People moved slowly then. They ambled across the square, shuffled in and out of the stores around it, took their time about everything" (Lee 5-6).
Since Atticus recognizes the scope of his battle, he focuses on one aspect with higher odds for success: positively influencing his own kids. If he raises Scout and Jem to understand the nature of injustice, then they will be in a better position to stand against it as they grow up.