Prospero's master-servant relationship with the spirit Ariel begins when Prospero releases Ariel from inside a pine tree where Ariel has been imprisoned by "This damned witch Sycorax" (1.2.314) for refusing "To act her earthy and abhorred commands" (1.2.324). Ariel remained inside the tree for twelve years, until Prospero came to the island.

PROSPERO. It was mine art,
When I arrived and heard thee, that made gape
The pine and let thee out. (1.2.343–345)

Prospero angrily reminds Ariel of this fact and calls Ariel a "malignant thing" (1.2.306) and his "slave" (1.2.321) when Ariel asks Prospero to honor a promise he made to Ariel to release him from Prospero's service in one year.

This is the only time in the play that Prospero refers to Arial as "my slave." In this same scene, before Ariel reminds Prospero of his promise, Prospero calls Arial "servant," (1.2.219), "My Ariel," (1.2.221), "My brave spirit" (1.2.241), and "my spirit" (1.2.252), all of which denote servitude to Prospero—which is also reflected in Prospero's use of the possessive "my" when referring to Ariel throughout the play—or even a friendly master-servant relationship; but none of those endearments implies slavery.

It's only when Ariel angers Prospero that Prospero refers to their relationship on a baser level and threatens to imprison Ariel in an oak tree for another twelve years. This might be an idle threat, because Prospero has no powers over nature, even though he claims that he released Ariel from inside the tree with his "art" (1.2.343), but it's a threat that's sufficient to keep Ariel subservient to him.

Prospero has no reason to doubt Ariel's loyalty to him. Ariel never refuses to do what Prospero asks him to do, "be't to fly / To swim, to dive into the fire, to ride / On the curled clouds," (1.2.223-225), or to cause the raging tempest at sea that opens the play. Ariel always performs his tasks to the letter, and Prospero invariably takes credit for Ariel's work.

For example, Prospero implies to his daughter, Miranda, that he caused the tempest and takes credit for it, but, in fact, Ariel did it all, all by himself.

PROSPERO. Hast thou, spirit,
Performed to point the tempest that I bade thee?

ARIEL. To every article. (1.2.227–229)

It's important for Prospero to reinforce his domination over Ariel and perhaps even to claim ownership of Ariel because Prospero needs Ariel. Ariel has skills that Prospero needs to implement his magic—if, indeed, Prospero actually has any powers of his own—and Ariel has powers, particularly powers over the natural world, that Prospero needs but doesn't possess.

Prospero exerts his possession and control of Ariel throughout the play, even after Prospero divests himself of his own magical powers, because Prospero has need of Ariel for one more task. Prospero promises Alonso, King of Naples, and the others with whom Prospero is returning to Naples, that he, Prospero himself, will deliver "calm seas, auspicious gales" (5.1.365).

Prospero then immediately turns to Ariel to have Ariel deliver on Prospero's promise to ensure his ship's safe passage to Naples—for which Prospero will no doubt take full credit—after which Ariel will be free.

PROSPERO. My Ariel, chick,
That is thy charge. Then to the elements
Be free, and fare thou well! (5.1.367–369)

This is when Prospero finally grants Ariel his freedom: Prospero no longer needs Ariel, or Ariel's powers, to do his bidding.

Prospero, when stranded long ago on an island with his toddler daughter, encountered several beings that already lived there. The two part-human, part-supernatural beings were Ariel and Caliban; the genders of both beings are open to interpretation (though Ariel is referred to using he/him pronouns twice in the play). Prospero acts as a colonial ruler and seems to consider himself a benevolent patriarch. He finds it important to dominate and manipulate everyone, by his wits when possible and by magic otherwise.

The exact hold that Prospero has over Ariel, while not precisely stated, is presumably at least part magic. Ariel rarely resists outright but tries to placate Prospero. Again, the exact reasons for Ariel’s deferential behavior are not spelled out. It seems that Ariel accepts being bound in a servile relationship to the master Prospero, one that was imposed when Prospero freed the sprite from being trapped in a tree trunk. It is also likely that Prospero’s magic may be strong enough to prevent any truly independent actions. While Ariel has some supernatural powers, especially over the elements (as shown by conjuring a storm), their relative strength compared to Prospero’s powers remains debatable. Ariel can change shape, becoming fire during the storm and assuming the shape of a harpy—things we never hear of Prospero doing. As it was Caliban’s mother who previously bewitched Ariel into the tree, Ariel may need Prospero’s protection against Caliban, who is also in servitude to the magician.

Prospero, living in exile, is strongly focused on getting his daughter, Miranda, into a position of authority over his former lands, which he intends to accomplish by marrying her off to Alonso, the son of the King of Naples. To carry out this plan (of which he keeps her in the dark), he needs the assistance of other island dwellers. Although he did not know of Ariel’s powers when he freed him from Sycorax’s magic, he has learned something of the sprite’s abilities and, before the play’s action began, had employed Ariel’s power to create the storm that shipwrecked Ferdinand and Alonso. The question of who is serving whom, as Prospero is dependent on Ariel to make his plot work, arise constantly. From act I, scene 2, when Ariel reminds Prospero of his promise, Shakespeare seems to suggest that Ariel is more honest and honorable than his human master.

In "The Tempest," the relationship between Prospero and Ariel is one of master and servant.  Prospero is the master and Ariel is the servant.  In Act 1, Scene 2, Prospero calls to Ariel, "Come away, servant, come."  Then Ariel greets Prospero by saying, "All hail, great master!"

Despite the fact that their conversation is friendly, it appears that Ariel wants his freedom.  Later in the same scene Ariel feels he must remind Prospero of his promises, "Let me remember thee what thou hast promised... my liberty."  Prospero is offended at Ariel's reminders and decides that Ariel also needs some reminding.  Prospero goes into a lengthy discussion, reminding Ariel how how he was saved from imprisonment.  He finishes by threatening Ariel, 

If thou more murmur'st, I will rend an oak

And peg thee in his knotty entrails till

Thou has howled away twelve winters.

Ariel remembers his place as Prospero's servant and answers, "Pardon, master."

The relationship between Prospero and Ariel is one similar to master and slave.  When Prospero arrived on the island, Ariel had been imprisoned in a tree by the witch Sycorax who ruled the island.  Apparently, Prospero somehow defeated the witch and was able to free Ariel from the tree.  Prospero then claimed that Ariel owed him a debt of service for freeing him from his imprisonment.  Ariel agrees to do magic for Prospero, and it seems as if there had been a set length of time for Ariel's service.  During the play, Ariel tells Prospero that he always promises to set him free, yet he never lives up to the promise.  Prospero tells Ariel that he will set him free after he performs one last act of magic.  It is only after Prospero creates the magic circle and restores order among himself and Alonso and the other men that he sets Ariel free.