What are Danforth's conflicts with Hale, society, and within himself in The Crucible?

Part of Danforth's conflict with Hale is that Hale is making suggestions about how the court should proceed: he believes, for example, that John Proctor and Giles Corey should be allowed to leave and return with a lawyer to help them present their evidence.  However, Danforth explains why a lawyer seems totally unnecessary to him: the only people who can be aware of witchcraft are the witch and her victim, and since the witch will never accuse herself, the only person who the court needs to hear from is the victim -- what could a lawyer contribute in such a case? he asks.

Also, Hale defends Elizabeth and John Proctor when Danforth tests them in order to try to get to the truth about why Elizabeth dismissed Abigail from their service.  Elizabeth tells a "natural lie," according to Hale, as she attempts to protect her husband, but Danforth will not listen and chooses to believe that he's proven that Abigail is innocent of wrongdoing.  Hale, however, claims that "private vengeance is working through this testimony" and that he can no longer "shut [his] conscience to it [...]" anymore.  Danforth disagrees.

Further, Hale is skeptical of the "evidence" presented by Abigail against Mary Warren -- the testimony about the yellow bird that is supposedly Mary's spirit.  Hale tries to serve as a voice of reason during this chaos, saying that he can see nothing of what Abigail describes, and he tries to object, but Danforth shuts him down, saying, "I will have nothing from you, Mr. Hale."  Danforth takes issue with Hale's interruptions and insistence that Abigail and the girls are guilty of the attempted murder of those they accuse; Hale believes Mary Warren's testimony, but Danforth believes that he has disproven any guilt relating to Abigail.  Danforth believes there are really witches in Salem, while Hale seems to have arrived at the truth: that the girls are making it all up.

Later, in Act Four, after Hale has returned to Salem, Danforth takes issue with Hale's goals.  Hale says that he counsels the innocent persons condemned to die to lie and confess in order to save their own lives.  Judge Hathorne declares that "It is no lie," and Hale exclaims, "It is a lie!  They are innocent!"  However, Danforth will "hear no more of that!"  He is clearly disgusted with Hale at this point, and when Hale next tries to speak to him, he "press[es] Hale impatiently aside" and eventually "brushes past Hale and walks out," ignoring Hale's attempts to talk with him.

What is Danforth's internal conflict in The Crucible?

Deputy Governor Danforth's internal conflict concerns his decision to continue sentencing innocent individuals or immediately put an end to the corrupt court and witch trials. During act 3, Danforth is faced with damning evidence from Giles Cory and John Proctor that Abigail and her followers are falsely accusing innocent citizens. In act 4, Reverend Parris informs Danforth that Abigail and Mercy Lewis have fled the town, which confirms that they have been fabricating their testimonies. Danforth is now faced with the dilemma of whether to admit that the court's integrity has been compromised or remain resolute and follow through with the executions. Unfortunately, Danforth remains obstinate and refuses to acknowledge his mistakes. As a religious man, Danforth is also struggling with his faith and must realize that the fate of his soul hangs in the balance. Rather than admit that the court is corrupt and he is responsible for the deaths of numerous innocent citizens, Danforth refuses to postpone the executions or appear weak. He tells Reverend Hale in act 4,

"Twelve are already executed; the names of these seven are given out, and the village expects to see them die this morning. Postponement now speaks a floundering on my part; reprieve or pardon must cast doubt upon the guilt of them that died till now. While I speak God’s law, I will not crack its voice with whimpering. If retaliation is your fear, know this - I should hang ten thousand that dared to rise against the law, and an ocean of salt tears could not melt the resolution of the statutes" (Miller, 131).