The majority of conflicts in Shakespeare's The Tempest swirl around Prospero, former Duke of Milan, whose brother, Antonio, usurped his dukedom and set Prospero and his daughter, Miranda, adrift on the sea. This is how Prospero and Miranda came to be marooned on the island on which they've lived for twelve years, which Prospero rules with magic and spells.
The major conflict is between Prospero and Antonio, who was aided in his overthrow of Prospero by Alonso, the King of Naples.
Prospero causes a violent storm which shipwrecks Antonio, Alonso, and others on Prospero's island. Prospero intends to effect some kind of revenge against Antonio and Alonso, although Prospero's not quite sure what form that ultimate revenge will take, except to punish Antonio and Alonso and restore himself as Duke of Milan.
A secondary conflict involves Prospero and Caliban, an ill-tempered son of Sycorax, a witch. Prospero enslaved Caliban when Prospero and Miranda were first marooned on the island.
Prior to Prospero's arrival, Caliban ruled the island, but he was overcome by Prospero's magic, and Caliban now plots against Prospero with Alonso's jester, Trinculo, and buffoonish Stephano, who were shipwrecked on the island with Antonio and Alonso.
Prospero is also involved in a lesser conflict with Ariel, a kindly spirit who Prospero also enslaved when he came to the island. Prospero takes advantage of Ariel's good nature and repeatedly fails to keep his word to free Ariel from his servitude to Prospero.
A minor conflict arises between Prospero and Miranda regarding Ferdinand, Alonso's son, with whom Miranda has fallen in love at first sight. Prospero calls him a traitor and mistreats him with magic spells, for which misdeeds Miranda calls Prospero to task.
One conflict to which Prospero isn't a party involves Antonio and Alonso's brother, Sebastian, who conspire to kill Alonso and set up Sebastian as the king of Naples.