In Ch. 17, Atticus is trying to prove that Tom Robinson did not beat Mayella. Mayella and Bob are claiming that Tom Robinson beat and raped Mayella. She says that she had bruising on the right side of her face, meaning her attacker would have to be left-handed.
Atticus asks Bob to write his name. Bob writes with his left hand, meaning he could have left the bruises on Mayella. On the other hand, Tom Robinson is right handed, but not by choice; he cannot use his left hand at all because it was caught in a cotton gin and is now unusable.
This evidence casts the shadow of a doubt on Bob and Mayella's testimony that Tom was the one who beat Mayella. Atticus is trying to prove that Tom may be innocent. He succeeds in causing the audience (both the jury and the reader) to doubt the Ewells' word. This makes the eventual conviction that much more frustrating. Despite these doubts, the jury still believes that Tom was guilty. This clearly illustrates the racism that pervades Maycomb.
"Atticus had used every tool available to free men to save Tom Robinson, but in the secret courts of men's hearts Atticus had no case. Tom was a dead man the minute Mayella Ewell opened her mouth and screamed." (Chapter 25)