Atticus's definition of trash is a white man who takes advantage of a black man. He sees this as morally wrong, because the one in power is exploiting the one without power. If anything, the one in power should protect the one without power. Atticus makes this point in a conversation with Jem. Here is what he says:
As you grow older, you’ll see white men cheat black men every day of your life, but let me tell you something and don’t you forget it— whenever a white man does that to a black man, no matter who he is, how rich he is, or how fine a family he comes from, that white man is trash.”
A person who would fit Atticus's definition of trash would be Bob Ewell.
Aunt Alexandra's understanding of trash is quite different. Her definition does not include morality. What is more important to her is the pedigree and sophistication of a person. Therefore Aunt Alexandra does not want Scout to play with Walter Cunningham. To her, the Cunninghams are trash.
She took off her glasses and stared at me. “I’ll tell you why,” she said. “Because— he—is—trash, that’s why you can’t play with him. I’ll not have you around him, picking up his habits and learning Lord-knows-what. You’re enough of a problem to your father as it is.”
As the book progresses, Mr. Cunningham proves to be a honorable man. Atticus says that he is the only juror who is standing up for Tom Robinson. This shows that Alexandra's understanding of what trash is is flawed.