What is the relationship between Ariel and Prospero in Act I, Scene 2 lines 189 to 321?
The relationship is largely one of servant and master. If you look at opening of the exchange, Ariel starts with "All hail, great master!" After that, many expressions of servitude and inferiority follow.
For his part, Prospero seems to take his right to command for granted.
Near the end of the scene, Ariel does show some independent spirit, some signs that things were not always this way and will change again.
The relationship starts of as Master and servant (with a hint of friendship) , as Ariel is usually calling him master, but as this scene goes on it turns from Master and Servant to Master and Slave. Note how prospero says "Thou liest malignant thing!" and this basically mean Prospero is calling Ariel a dirty liar. You would not usually insult your servant in such a way, but more of your slave, so on this basis is why I say it is Master and Servant, and Master and Slave.