A symbol in a piece of literature is a recurring object, color, etc., that is used to support a major theme in that piece of literature.
To understand the significance of Tom Robinson representing the symbol of a mockingbird, one needs to recall Atticus' instructions to the children when they received their air rifles as gifts, and Miss Maudie's explanation:
“Remember it’s a sin to kill a mockingbird.” That was the only time I ever heard Atticus say it was a sin to do something, and I asked Miss Maudie about it.
“Your father’s right,” she said. “Mockingbirds don’t do one thing but make music for us to enjoy . . . but sing their hearts out for us. That’s why it’s a sin to kill a mockingbird.”
The overall concept is that it is ethically wrong to harm anything or anyone that does nothing to harm others—it's wrong to harm an innocent.
There are a couple of characters in Harper Lee's To Kill a Mockingbird that can be seen as mockingbirds. One in particular is Tom Robinson, a poor black man, and a father and husband. He works very hard to support his family. Mr. Dees (Tom's employer) speaks highly of the dedicated worker Tom is. Tom is a good man at heart, as well: he is polite and kind.
Tom Robinson does nothing to bring harm anyone, but is punished because he shows the kindness toward Mayella Ewell—innocently, he offers to help her do work in her home. She makes a pass at him and Tom desperately tries to remove himself from the situation. However, when Bob Ewell comes upon them, he beats Mayella and she cries "rape." Had it been a white man, Ewell would probably have beaten him up. However, because of Tom's race, Ewell's prejudice, and the fact that Tom (a black man) feels sorry for Mayella (a white woman), Tom never has a chance of survival.
In Chapter 19, Tom described the sympathy he felt for Mayella: a young woman who worked so hard without anyone to help her. Mayella often asked Tom when he passed by from work each day for help around the house.
Atticus asked Tom:
Were you paid for your services?
No suh, not after she offered me a nickel the first time. I was glad to do it, Mr. Ewell didn't seem to help her none, and neither did the chillun, and I knowed she didn't have no nickels to spare.
The fact that Tom is falsely accused of rape while trying to help Mayella shows how he is like a mockingbird: doing no harm but being harmed.
In Chapter 24, Atticus describes the excessive use of force to stop a man with only one functioning arm; this also shows how Tom can be seen as a mockingbird:
...the guards called to him to stop. They fired a few shots in the air, then to kill. They got him just as he went over the fence...Seventeen bullet holes in him. They didn't have to shoot him that much. [...] ...what was one Negro, more or less, among two hundred of 'em? He wasn't Tom to them, he was an escaping prisoner.
Mr. Underwood, who runs The Maycomb Tribune, the town's newspaper, most clearly reflects the sin in the senseless killing of Tom Robinson, found in Chapter 25:
Mr. Underwood didn't talk about miscarriages of justice, he was writing so children could understand. Mr. Underwood simply figured it was a sin to kill cripples, be they standing, sitting or escaping. He likened Tom's death to the senseless slaughter of songbirds by hunters and children...
And the sacrificial essence of Tom Robinson, a black man, is evident with Scout's observation:
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.
In every detail concerning Tom Robinson, he is presented as one who (like a mockingbird) does no harm, but becomes the object of hatred and mistrust, and is killed not because he is guilty, but because he is black and believed to be incapable of innocence by virtue of his race.