The reaction to the dog is one of great concern and fear. The children first notice "Tim Johnson" behaving strangely and call on Calpurnia to see what she thinks might be wrong. Calpurnia immediately recognizes the signs of its being a "mad dog" and gets the children inside. She then calls Atticus for assistance.
Moments later, Atticus shows up with the sheriff, and they agree the dog's "got it" - rabies - and must be killed immediately before he can pass on this disease to others. He's right near the Radley house, though, and it's important the dog be killed with one shot - he could easily charge at someone otherwise, thus the situation is very dangerous.
It is here the children first learn of their father's hidden talent: shooting. He used to be called "ol' one shot Finch," and the sheriff asks Atticus to be the one to shoot the dog. He initially tries to get out of it, but then he takes on the responsibility and does what he has to in order to protect his children and the others in his town.
This act takes on symbolic significance in that the dog, just like the town's prejudice, presents a threat to Jem and Scout. In shooting the dog, Atticus protects them from a physical threat. In defending Tom and speaking and acting for what is right, Atticus tries to protect them mentally and emotionally; he hopes to prevent his children from "catching Maycomb's usual disease" of prejudice.
He didn't want to have to shoot the dog, and he'd rather not have to take this case - he's rather "let this cup pass" from him and avoid this burden. He takes on the case, though, because he knows it's the right thing to do, and if he didn't, he would feel like a hypocrite, unable to tell his kids "what to do" any more because he would know he wasn't following his own moral compass.