In the trial of Tom Robinson, Mr. Gilmer is the circuit solicitor. Scout describes him as a man who was from Abbottsville, whom she and Jem and Dill only saw when court was in session. He was
a balding, smooth-faced man, he could have been anywhere between forty and sixty....we knew he had a slight cast in one of his eyes which he used to his advantage: he seemed to be looking at a person when he was actually doing nothing of the kind, thus he was hell on juries and witnesses.
There is a tension between Atticus Finch and Mr. Gilmer that develops in this chapter because when Mr. Gilmer interviews Sherriff Tate, and Mr. Tate testifies that Mayella was "pretty well beat up," Mr. Gilmer just thanks Mr. Tate for his testimony, but Atticus asks why a doctor was not called. Mr. Gilmer's disinterest seems to irritate Atticus because he senses that Mr. Gilmer feels his case is already won.
Then, when Atticus asks Bob Ewell if he can write and read, Mr. Gilmer objects, wondering what influence "Mr. Ewell's education had on the case." As Atticus continues, Mr. Gilmer half-stands and half-sits at his table, actions that suggest his discomfiture with Atticus's questions which establish Mr. Ewell as being left-handed. For, Mr. Gilmer realizes that by pointing out that Bob Ewell is left-handed, Atticus can establish that Tom Robinson is not guilty of striking Mayella since the marks on her face were made by someone who has use of his left hand and arm. Knowing this crucial fact could sway even a complacent jury, Mr. Gilmer becomes angry
We are given the answer to your question early on in Chapter 17. This is a key chapter for you to focus on, so it would be well worth your while to read it and to appreciate how Harper Lee creates an amazingly tense court scene. We are told by Scout, the first person narrator of this excellent novel, that the solicitor is actually someone who comes from Abbottsville and is not well known to them and to Maycomb itself. However, Jem and Scout have met him enough times to know that he has a feature that helps him in his job as a solicitor. Note how he is presented:
The solicitor, a Mr. Gilmer, was not well known to us. He was from Abbottsville; we saw him only when court convened, and that rarely, for court was of no special interest to Jem and me. A balding, smooth-faced mad, he could have been anywhere between forty and sixty. Although his back was to us, we knew he had a slight cast in one of his eyes which he used to his advantage: he seemed to be looking at a person when he was actually doing nothing of the kind, thus he was hell on juries and witnesses.
The solicitor is therefore called Mr. Gilmer, and although he is not well-known in Maycomb and to the children, he certainly has a reputation and the children have met him enough times to know about how he uses the cast in his eye to his benefit.