One of the reasons The Tempest is considered a comedy is because Prospero forgives the men who have betrayed him. When do you think Prospero makes the decision to forgive them? Are there any characters that deserve an apology from Prospero?

Throughout the play, Prospero has been dictatorial and controlling. But in the final act he becomes forgiving. The play offers three reasons for this change.
First, Ariel motivates Prospero to forgive. Ariel reports to Prospero that his enemies are sorrowful, pointing out particularly that Gonzalo is crying so hard that tears run down his beard. Ariel says that if Prospero were to see them as Ariel has, Prospero would forgive them. Ariel gives the following description of them to his master:

The King,His brother and yours, abide all three distracted,And the remainder mourning over them, Brimful of sorrow and dismay; but chieflyHim that you termed, sir, the good old Lord Gonzalo.His tears run down his beard like winter's dropsFrom eaves of reeds. Your charm so strongly works 'emThat if you now beheld them, your affections Would become tender.

Prospero asks Ariel if he really thinks his heart would soften if he saw his enemies. Ariel doesn't say what he thinks Prospero would feel but instead states:

Mine would, sir, were I human.

By saying this, Ariel communicates that forgiveness is a human quality. This encourages Prospero to say his own heart will become forgiving. How can he, he says, not forgive his fellow human beings when a spirit, which is "but air," feels compassion towards them? Prospero thus declares that he cannot be less kind and forgiving than a nonhuman. Ariel, by modeling merciful feelings, shames Prospero into forgiveness. Prospero states:

Hast thou, which art but air, a touch, a feeling Of their afflictions, and shall not myself,One of their kind, that relish all as sharplyPassion as they, be kindlier moved than thou art?

A second reason Prospero offers for forgiveness is that it is the rarer, and hence nobler, path. We see in Prospero's forgiveness a man who wants to be thought well of by others. He will take the high road, and in so doing, set himself apart from most people:

The rarer action isIn virtue than in vengeance

Finally, Prospero forgives his brother Antonio for stealing his kingdom, because he knows Antonio must return it to him. Prospero says to Antonio:

For you, most wicked sir, whom to call brotherWould even infect my mouth, I do forgive Thy rankest fault, all of them, and requireMy dukedom of thee, which perforce I knowThou must restore.

This isn't the fullest form of forgiveness, as we note that Prospero still feels it would "infect" his mouth to call Antonio his brother. Obviously, he still has harsh feelings towards his sibling. It could also be called a transactional kind of forgiveness to forgive because you are going to get something in return. Nevertheless, Propero is making an attempt to do the right thing. He may still be angry at his brother, but he will no longer seek revenge against him. This is a step in the right direction.
However, Prospero never forgives Caliban. Caliban does deserve an apology. He helped Prospero when Prospero first shipwrecked on the island, teaching him to survive. Once Prospero learned this, rather than respond gratefully, he enslaved Caliban. Many critics have read this as colonialism in microcosm: whites coming to a foreign place, relying on the native dwellers to survive, and then betraying, harming, and enslaving them—and justifying doing so by labeling them subhuman. Caliban does strike back against Prospero, but one could argue that Caliban is more sinned against than sinning.