What does Atticus mean when he say's "the courts are the great levelers"?
This is a fairly ironic statement of Atticus'. He states many concepts in his closing argument as if they are true, when in reality they are SUPPOSED to be true.
This is also the case in the piece he brought up about Thomas Jefferson's words, "all men are created equal." Even when Jefferson wrote that, he had slaves and it wasn't a true statement. This shows how we rely upon the ideal, but do not necessarily live it out.
Atticus was a great speaker and this effort to state the ideal did at least influence one juror who held out for a while before agreeing to give a 'guilty' verdict. The purpose of a jury is to have a body of peers in judgment of the accused. These jurors were not level with or equal to Tom, they saw themselves as better. But one person, one person was beginning to move.
Atticus makes this statement in Chapter 20. When he does, he is making is final argument to the jury in the case of Tom Robinson.
What Atticus is saying is that courts are supposed to treat all people equally. No matter how rich or influential one person is, that person is no different (before the law) than a poor person. He uses the word "leveler" to mean that courts bring all people to the same level.
What he is trying to do here is influence the jury. He's telling them that American courts are made to treat people equally. He's asking them to give Tom fair treatment.