The Ewells can be considered "poor white trash" for a variety of reasons. Different characters in the book, though, would give different explanations for why they are "trash". Aunt Alexandra would say they are "trash" simply because they are dirty and poor (as shown when she tells Scout she cannot play with Walter), but others provide further explanation.
It is Scout who first explains the various stereotypes for the many families in Maycomb. According to her (and Jem), the Ewell family has been living out there by the garbage dump for as long as anyone can remember. They live in a run-down, over-crowded house, which is neglected and dirty, and the children run wild. The family has been receiving various forms of public assistance for a long time because no one holds down a steady job to earn money.
Looking at this generation (and we can assume earlier generations have been mostly the same), Mr. Ewell is an alcoholic who wastes what little money the family has on liquor. The town allows him to hunt out of season because otherwise the children would go hungry. He abuses his children physically, and we know based on what Tom said in his testimony that he abuses Mayella (and perhaps others) sexually as well. He truly is a despicable man.
The town contibutes, though, to this vicious cycle of poverty by not insisting that the children attend school. The law requires that they have their names on the school rosters, so they insist on this much and the kids come to the first day of school, but no one worries too much if they never show up again.
Through the behavior of Burris Ewell in Scout's class, we see that the children have not taught to be clean, well-mannered, or respectful. Scout describes Burris as the "filthiest human" she has even seen, and he seems pleased with himself when he makes the teacher cry after telling her off. When she sends him home because of his "cooties," or lice, it is a relief to everyone in the classroom. They don't want him there, so why bother insisting on his attendance? Except that, as a result, nothing ever changes for the Ewell family.
Mayella, Scout says, seems to perhaps try harder than the others. At the trial, she appears cleaner (not as freshly scrubbed as Mr. Ewell), and Scout recalls seeing geraniums planted at the Ewell house. It is not hard to understand why Mayella would fall in love with Tom - a handsome, hard-working man who treats her kindly and takes time to help her out. However, when he does not return her advances, and when her father sees what she is up to, both of them turn on Tom and blame him for things he has never done.
This, according to Atticus, makes them "trash," for he believes that anyone who would take advantage of another person, especially one beneath him in society, deserves this title. It has nothing, in his mind, to do with their being poor or even dirty. It has to do with their lack or integrity - or moral, ethical behavior. This is the lesson he wants to teach his children.