Irony is when an event turns out to be the opposite of what one would usually expect.

For example, in Act IV, Reverend Hale encourages Goody Proctor to lie to avoid being executed. Considering how the Puritans preach against sin and view willful wrongdoing as a one-way trip to hell, that a reverend should encourage people to lie to save their earthly lives is deeply ironic. It also shows how far from righteous the political and religious system in Salem has become.

Another example of irony is in Proctor's redemption. Proctor is an adulterer, the greatest sinner of anyone in the scene. And yet, he chooses to be wrongfully executed so his children will not be shamed by having a coward for a father and will receive their proper inheritance. In the simplistic world of the Puritans of the play, this might be even more ironic than for the audience, since such redemption in some ways goes against what they would expect.

An example of verbal irony is when Danforth claims that Elizabeth, not weeping over Proctor's impending hanging, claims she is cruel: "A very ape would weep at such calamity! Have the Devil dried up any tear of pity in you?" This is ironic since Elizabeth loves Proctor more than anyone else. She lied to save his good name when she was at risk of being executed as a witch and now, loves him more than ever as she is about to lose him to the gallows.

It is ironic that Mr. Hale, a minister, would counsel people to lie. We would most likely expect him to caution people not to lie, seeing as how lying is considered to be a sin, and yet, he returns to Salem to encourage those convicted and sentenced to hang to lie and confess to witchcraft in order to save their own lives. He says to Elizabeth Proctor,

Life, woman, life is God's most precious gift; no principle, however glorious, may justify the taking of it. I beg you, woman, prevail upon your husband to confess. Let him give his lie. Quail not before God's judgment in this, for it may well be God damns a liar less than he that throws his life away for pride.

Hale tries to get Goody Proctor to convince her husband, John, to confess a lie to save his life. This runs contrary to our typical expectations for a minister, and therefore it is ironic.

Secondly, it is ironic that Deputy Governor Danforth, the primary magistrate during these proceedings, cares less about achieving justice and more about his personal reputation. He is a judge in these trials and therefore his primary duty is to punish the guilty and exonerate the innocent. Instead, by refusing to postpone the hangings, he —seemingly knowingly—executes innocents simply because 

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.

Thus, we see that the judge doesn't care so much about justice: this is ironic. Danforth will not pursue the idea that these individuals are actually innocent, because that would mean that the others he already executed were likely innocent too. Such a revelation would completely undermine his authority, and he might be removed from power.

Thirdly, Marshal Herrick's response to Sarah Good and Tituba at the beginning of the scene is ironic as well. The two women talk about how the Devil is coming and he is going to give them wings so that they can fly to Barbados and escape Salem. Earlier in the play, such talk would have been taken quite seriously, as all accusations and "admissions" were. Now, however, no one pays them any mind, and this goes contrary to what we might expect.

Part of what make The Crucible so powerful is its use of irony.  Remember when looking for irony, you're looking for anything that is the opposite of what you expect.

In Act IV, Hale returns to Salem.  He left in disgust at the end of Act III, but his conscience has brought him back.  He tells Danforth

I have come to do the Devil's work. I have come to counsel Christians they should belie themselves.

Reverend Hale came to Salem with the belief that he was helping free the afflicted girls of withcraft; however, he has learned that it was his investigation which has actually lead these people to be condemed and many executed. Now he has returned to bring the characters "to God." But since these are already holy people (Martha Corey, Rebecca Nurse, and John Proctor) he is "bringing them to God" but getting them to lie and confess to sins they did not commit.

Proctor's confession is ironic.  He is able to rationalize to himself and Elizabeth that he is not an innocent man.  He has sinned before (Abigail) so what's wrong with confessing to this sin?  He doesn't see himself as a holy man anymore, so he feels he should be able to lie and live.  The irony occurs when he finds that he cannot lie and instead recants his confession.

The end of the play is ironic. Proctor finds peace with himself and his actions when he rips up his confession.  Of course, ripping up the confession is a sign of guilt in the eyes of the court, so he will be sent to the gallows, but as Elizabeth tells Reverend Hale, he has regained his goodness and sense of who he is.

He have his goodness now.  God forbid I should take it away from him!