The mob wants to lynch Tom Robinson in order to send a message to other men in the African-American community and to prevent Tom's going to trial.
Thought to be named after Charles Lynch, a planter from Virginia who held an "irregular court" intended to punish loyalists to the British, lynching was used to enforce both formal laws and numerous unwritten rules of conduct meant to assert white domination. So, when the lynch mob comes for Tom Robinson, they attempt to rid their town of any controversy that a ruling in court might cause. More than any other reason, they wish to maintain racial control through fear.
Whether Tom is guilty or not may be unimportant to the lynch mob. The mob simply wishes to maintain the position of control by any means. Tom threatens the established order of Maycomb, an order that keeps black people in lowly positions politically and socially. Hanging Tom intimidate others. These are the reasons they tell Atticus to "stand aside" in chapter 16.
The mob wants to lynch Tom because to them, Tom represents a threat to their segregated way of life. Tom has been arrested for raping Mayella Ewell, the worst thing a Black man could do. Rape of a white woman by a black man challenges the idea of segregation and separation of the races at a very basic level. The mob opposes Tom both because he is black and also for his supposed actions. However, because of Scout's actions of singling out Walter Cunningham, the reader sees that this mob is really composed of people who would act differently as individuals. Scout realizes this, too, and is able to defeat the cowardly ways of the mob by reminding them of their individuality and responsibility.
The mob has three things against Tom: 1) Tom is black and they are white in the very racially charged south 2) Tom is accused of beating and raping Mayella Ewell, a white woman 3) they are angry at Tom for being a decent human being who suggested through his testimony that none of the white people were helping Mayella--that her life was difficult and Tom felt sorry for her. This statement enrages the crowd as Tom, a black man, should not be feeling superior or sorry for anyone white--not even Mayella Ewell, the daughter of the white trash in the community.
the mob wants to lynch Tom because they think that he raped Myella Ewell.