The Tempest starts with an actual storm, or tempest, created by Ariel to bring ashore to the island the men who helped Antonio usurp the throne that belongs rightfully to Prospero. The imagery of the ship in the storm is of a vessel that the high seas and violent winds have torn asunder. As the ship is about to go down in Act I, scene 1, voices cry out:

"Mercy on us!—We split, we split!—Farewell, my wife and children!—Farewell, brother!—We split, we split, we split!" (lines 36-38)

The image of the ship being torn apart has a symbolic meaning as well, as the ship of state (referring to the government) is about to be torn apart too. Antonio, who has wrongfully usurped the dukedom, is eventually to be unseated in favor of Prospero, the rightful duke. The storm is a symbol for the coming dissolution of the current political structure. 

The significance of The Tempest in Shakespeare's time is that it was written in 1610, a year after the British first colonized Virginia. A ship called the Sea Venture was wrecked during a storm off Bermuda in 1609 during the third supply mission to the new colony of Virginia, and its wreck is believed to have inspired Shakespeare to write The Tempest. 

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.