The author chose that title because one of the major themes of the story is that people should not abuse or kill things that do not do any harm. A mockingbird is something that harms no one and so you should not harm it.
There are a few main plot lines. First, Jem and Scout Finch are very interested in finding out more about Boo Radley who has been locked up in his house for more than a decade. They start out thinking he's a monster but they learn that he is human as the story goes on.
The other main plot is that their father, Atticus, defends Tom Robinson, a black man accused of raping a white woman. The plot follows this trial as Robinson is unjustly convicted.
Both Radley and Robinson are "mockingbirds" who are abused or killed by the people of the town without good reason.