When I first stared reading The Tempest I was intrigued by Ariel - after I got over continually thinking about the little mermaid. So I thought I'd do a little research about the supernatural in Shakespeare's plays.
The first plays I think of as to supernatural elements are The Tempest, and A Mid-Summer Night's Dream, or Even Macbeth and the witches. These three plays have magical characters, but they are also very different and work in very different ways. The fairies in a mid-summer night's dream are silly and naughty. Ariel on the other hand seems much more powerful and dangerous.
One thing I think is extremely interesting about Ariel is that he is trapped. He is kind of indentured to Prospero. But why the magical/mystical? In Macbeth Shakespeare wrote the witches in because King James was obsessed with demonology, going so far as to even write a book about it.
Scholar's don't seem to really know where the inspiration for Ariel and his fellow spirits came from. Some argue that he is a demon like in Macbeth, but others argue that he is more of a sprite. Religiously, spirits were either working for God or the devil. Scientifically they were only viewed as neutral essences that could be controlled by a person. - Pretty drastic differences.
Shakespeare writes about Ariel's powers frequently, as when Gonzalo cries, "I prophesied, if a gallows were on land, this fellow could not drown." Or gives examples, as when Ariel hits his hand against the table and the food laid out upon it 'magically' disappears.
Sycorax, the witch which tried to control Ariel before Prospero, adds another layer to the discussion. She is a witch a, worshiper of devils. Where she is often described in destructive terms, Prospero's use of Ariel's magic is described as wondrous. Because Sycorax is not able to control Ariel she locks him up in a tree, her desires for evil too strong for Ariel to obey. Strangely enough, Prospero is able to command Ariel because he is good.
Finally, at the end of the play, Prospero releases Ariel.