This title can be understood on both a literal and symbolic level. First of all, the literal definition of a "tempest" is a violent disturbance, sometimes related to weather, sometimes related to other upsetting commotion.
As the play begins, we meet the magician Prospero (ex-Duke of Milan), his lovely daughter, Miranda, the spirit of the winds, Ariel, as well as other spirits who live on a tropical, enchanted island. Prospero and the spirits battle a ship at sea transporting enemies. The enemies scatter over the island. The Price of Naples, Ferdinand, meets Miranda and falls in love with the handsome prince.
Thus, the tempest is literally the storm, and symbolically the disruption of the enchanted life of seclusion that Prospero has enjoyed.
As for the "effect of the era," Shakesperian scholar Stephen Greenblatt says the "Shakesperare's contemporaries were fascinated by the figure of the "magus" the great magician who by dint of deep learning, ascetic discipline and patient skill could command the secret forces of the natural and supernatural world." Though feelings about the occult were mixed, people were interested. Therefore, it is not surprising that characters like Prospero and Ariel crop up in the Renaissance.