In To Kill a Mockingbird, what did Mr. Heck Tate's mob want?
I would say that what they really want is to maintain status quo (the state of keeping everything the same). Tom Robinson has offended their sense of the way the world should work. In their racist views, blacks should be subservient and obedient and most importantly, invisible. Tom has offended that by not only feeling sorry for a white girl, but (in their minds), raping her. If this were about justice, the mob would want to see him go through the misery of a trial. If this were about protecting Mayella, the town would have acted years ago because everyone in that town knew that she was at risk from a drunken father.
No, that mob gets together because they want their town to go back the way it was with the blacks disappearing into the background, only appearing to clean up after the whites. They want to believe that in their world, whites will always defend whites because that's what's always happened in the past. Blacks were denied justice because the whites formed a united front against them. That's the comfortable, familiar world these racists live in. Only now, Tom and Atticus are threatening that world. These two men are making the racists uncomfortable because they're doing something unfamiliar and new. If the mob can reduce the "visibility" by either removing Tom and lynching him or by getting Atticus to give up the case, then they have achieved their goal.
There might be two answers, depending on which angry mob you are referring to. A group of men led by Heck Tate showed up at Atticus' house one evening demanding to know what his intentions are regarding the defense of Tom Robinson. Atticus informs them that he intends on defending the young man. Another angry mob (with worse intentions) shows up outside the jail, hoping to get their hands on Tom Robinson. Atticus is prepared for this reaction and with the help of Scout, the crowd disperses. They were intent on their own form of justice...