Discuss the significance of the title of the play The Tempest.

A tempest is a violent storm and considering that the first scene of the play takes place in such a storm, the title is quite fitting. It is this tempest, caused by Prospero's magic, that results in the entire plot and action of the play.
There is a symbolic meaning to the play's title as well. The relationships between many of the characters are as chaotic as a violent storm. Nothing goes smoothly for anyone, even the would-be puppet-master Prospero. He has to endure a mutiny that nearly throws all his plans awry.
Antonio's true motivations are laid plain to the King of Naples. The King for his part is distraught over the feared loss of his son. Almost nobody respects Gonzalo, despite his wisdom and optimistic outlook. Throughout all of this, we see Prospero's extreme anger towards his brother building before eventually subsiding as a storm does.
As we can see, the literal tempest exposed a the underlying stormy relationships between nearly every character in the play. In this sense, it is a very appropriate title.

In The Tempest, the final work that Shakespeare completed on his own, a sorcerer named Prospero hatches a plot to use his illusory powers to restore Miranda, his daughter, to her proper place. By creating a storm, he forces his deceitful brother to come under the illusion that they are shipwrecked on an island where the snake-like and conniving nature of Prospero's brother Antonio will be revealed. As a consequence, Prospero's daughter will marry the king's son Ferdinand.
The definition of tempest, according to Merriam-Webster, is "a violent storm." Thus, The Tempest refers, quite literally, to Prospero's storm that shipwrecks his brother and the king. On a deeper, more literary note, however, the titular tempest refers to the tumultuous plot of the play. With many different intersecting story lines, The Tempest, like most of Shakespeare's work, is structurally and literarily complex. While Prospero's scheme originally is seemingly simple, the tempest he conjures up in order to shipwreck Antonio ultimately creates a metaphorical tempest. Meanwhile, while the King and Antonio are shipwrecked, two drunkards attempt to create a rebellion against Prospero, Prospero attempts to play matchmaker with his daughter and the King's son, and Antonio and the King's brother plan to murder the King in order to commandeer power.
The Tempest contains two tempests: a literal storm, and a metaphorical tempest that refers to the chaotic plot that is set in motion.
https://www.merriam-webster.com/dictionary/tempest

The title of The Tempest has both literal and metaphorical significance. Literally, it refers to the storm, or tempest, whipped up by Prospero's magic to shipwreck the King of Naples and his crew. As storms go, it is quite torrid, but not fatal. Prospero might be a magician who is deeply embittered by his treatment, but he is not an evil man. He wants to make a point to those responsible for his banishment.
Just as a literal tempest brings the King of Naples to the island, so a metaphorical tempest originally brought Prospero there twelve years ago. Prospero had been Duke of Milan until he was overthrown in a palace coup by his sneaky brother, then cast out into exile. This sudden, rapid handover of power was itself a political storm, one that will have profound repercussions as the play progresses.