In To Kill a Mockingbird the character of Bob Ewell represents everything that is wrong with 1930s Alabama. A broken system of justice that is anything but blinded to color, gender, and culture, allows for fiends like Ewell to get away with a lot.
As a social parasite, he already demonstrates cowardice by allowing his entire family go awry, by taking relief checks and still living in a trash-filled environment in Maycomb, and by playing the system in using an innocent black man, Tom Robinson, as a scapegoat to make up for the indiscretions of his daughter.
Rather than owning up to what was obviously an attempt by his daughter to seduce Tom Robinson, Ewell accuses the man of rape and battery and carries on with this lie all the way to the courtroom. He wins, actually, precisely because the system is broken. However, rather than just leaving the matter alone, he embarks on a rampage of dishonorable acts that clearly show the type of creature that he really is.
First, he bullies Helen Robinson who is the widow of Tom Robinson. Thanks to Link Deas, who ran him off his property, that ended. Then, Ewell spits on Atticus's face, insults him, and invites him to fight. All of this bizarre behavior stemmed from the fact that, even though Ewell won the trial against Robinson, his reputation and that of his family worsened. The Ewells, who already were notorious in their community, now lost all credibility and respect. The only thing that an ignorant man like Ewell would choose to do as a result of this, is to go after the man who showed his (Ewell's) true colors.
While an invitation to fight may look at first glance like a "courageous" thing to do, let us remember that one is only as good as one's rival. It was Atticus who showed the class and courage that Ewell lacked by refusing to drop to Ewell's level. Atticus was even more courageous when he explained that he was "too old" for a fight of that nature. This adds further insult to the ridiculous behavior of Ewell.
Yet the ultimate show of cowardice for Ewell is when he goes after Jem and Scout and tries to kill them. Why go after two innocent children if his real issue was with Atticus? It is no wonder that the reader feels a cathartic relief when Ewell dies in the attempt.