Magic

A belief in magic and witchcraft was fairly standard in Shakespeare's day among both the educated and lower classes, and it features in several of his plays (most notably Macbeth, Henry VI, and The Tempest, but also Julius Caesar and Hamlet). 

"In 1583 Howard, Earl of Northampton, published his Defensative against the Poyson of Supposed Prophecies and in 1584 Reginald Scot presented Discoverie of Witchcraft in which, with great learning and ability, he exposed the pretensions of the magicians and their craft. He made many enemies by it; and James I ordered all the copies of it that could be found to be burned by the public hangman. In 1603 the king published his own book, Daemonologie, in the preface to which he asserts that he wrote the book "chiefly against the damnable opinions of Wierus 1 and Scot;" Wierus refers to John Van Weir, a Dutch physician who wrote in denial of the reality of witchcraft, and Scot apparently had different ideas about the origins and effects of magic than the king.  ("Magic, Books, and the Supernatural in Shakespeare's Tempest," Shakespeare's Comedy of The Tempest. Ed. William J. Rolfe. New York: American Book Company, 1904.)

It is believed Shakespeare wrote the weird sisters into Macbeth because he knew it would be attractive to James I, who was fascinated by magic and witchcraft.

In The Tempest, Prospero has many qualities understood by the Elizabethan/Jacobean everyman to be associated with magic practitioners:

In Macbeth, the weird sisters' process if described in gristly detail, but in The Tempest the spells and incantations are only hinted at: "my charms crack not," "my spirits obey," "untie the spell," etc. Perhaps that makes the magic more palatable to the audience.

It should be noted that Prospero's use of magic is not really maligned in the text.  However, Sycorax and other female magic users (the witches in Macbeth, Eleanor, Duchess of Gloucester in Henry VI Part 2) are repulsive, dangerous, and clearly evil.  This is evidence of the misogyny of the time; men practicing magic are clearly intelligent and forceful, but as King James I wrote in Daemonologie, women were more likely to become witches because they were “frailer” than men, and more prone to fall under the Devil's influence.  "Prospero belongs to the higher order of magicians — those who commanded the services of superior intelligences — in distinction from those who, by a league made with Satan, submitted to be his instruments, paying for the enjoyment of the supernatural power thus gained the price of their souls' salvation. The former class of magicians, as Scot remarks, "professed an art which some fond [foolish] divines affirm to be more honest and lawful than necromancy, wherein they work by good angels." Thus we find Prospero exercising his power over elves and goblins through the medium of Ariel, a spirit "too delicate to act the abhorr'd commands" of the foul witch Sycorax, but who answered his best pleasure and obeyed his "strong bidding." Over all this spirit world Prospero bears sovereign rule by the power of a commanding intellect. His subjects are "weak masters," he says; that is, weak individually, weak in the capacity for combining to make the most of their ability to do certain things that men cannot do. Prospero knows how to make them work in carrying out his far-reaching plans. "By your aid" he says, "weak masters, though ye be," I have wrought the marvels of my art" (Rolfe, 1904). 

It seems public opinion could have gone either way when seeing a portrayal of magic onstage. It might have been daring to depict the supernatural, but Shakespeare has Prospero renounce his magic at the end, bringing all characters to a acceptable place in the end.