Judge Danforth is primarily developed through indirect characterization. Unlike the characters introduced earlier in the play, we do not have the benefit of a narrator's explanation of Danforth and his behavior. Therefore, we primarily get to know Danforth by what he says and by his relationship with other characters.
Through Danforth's actions in the play, we understand him to be a very proud and stubborn man. He greatly dislikes and mistrusts Reverend Hale because he questions the proceedings. He also greatly dislikes Reverend Parris, calling him a "brainless man" (Act 4) and repeatedly telling him to be quiet (Act 3). Both of these men are challenging Danforth's authority in some way -- Hale by questioning the veracity of the girls' testimony and Parris by trying to take over the questioning of Mary Warren (see Act 3).
John and Abigail pose a different type of threat to Danforth. They question is judgment and discernment. Through these two characters we see the extent of Danforth's ego and stubbornness. The evidence against Abigail is staggering, but admitting that she was a fraud means admitting that he made a mistake in condemning supposed witches to their deaths, which Danforth cannot do.
These traits of Danforth seal the fates of the main characters. They cause Reverend Hale to compromise his beliefs, asking the prisoners to save their lives by lying. They lead to the condemnation of John Proctor, and allow Abigail Williams to escape her deserved fate. They even cause Reverend Parris, the court's biggest supporter, to plead and pray with prisoners to save themselves. Danforth's pride is the catalyst for the climax and resolution of the play.