The ultimate symbolic meaning of the gun seems to relate to notions of maturity and restraint. These concepts are directly aligned with each appearance of guns in the novel.
In the episode where Atticus shoots the rabid dog, we learn that Atticus was once the "best shot" in Maycomb. When he grew up, however, he put away thoughts of shooting.
Jem and Scout wonder why this would be. Miss Maudie explains that Atticus must have realized that his ability to shoot was a gift that other creatures did not share, which gave him an unfair advantage. In his sense of integrity and fairness, Atticus decided to give up the use of this skill - a decision expressive of his maturity.
Scout admires Atticus for his shooting talent, but Jem admires him for his gentlemanly restraint.
Scout and Jem, on the contrary, are excited to receive pellet guns for gifts. They are invigorated by the power that guns bring. This is expressive of their lack of maturity. Yet, they do learn restraint.
Immediately upon receiving the pellet guns, Scout and Jem are given an instruction about what they can and cannot shoot, forced to set limits on the use of their power.
Atticus tells them it's all right to shoot at blue jays, but "it's a sin to kill a mockingbird."
Restraint is also clearly at issue in the scene that takes place outside of the jail the night before the trial. Guns appear in this scene, but the power of restraint prevails.