Prospero's Books
Prospero's books are an important symbol of his power. Without them – as Caliban notes – he is vulnerable and no better than Caliban himself. His books also symbolize his greatest interest and passion (magic), which he chose to prioritize over his dukedom when he was still Duke of Milan. Many have compared Prospero to the figure of the playwright, who is so consumed by their artistry that they disregard other responsibilities and obligations.
The Tempest
The storm that opens the play is an important symbol of Prospero's anger and bitterness. He conjures the storm in order to seek revenge on those who wronged him, but it is important to note that the storm does not actually cause anyone physical harm. Prospero is controlling and vengeful, but he is not evil or murderous. Furthermore, the storm also represents a destruction of hierarchy (crucial to the play's backstory, in which Prospero was usurped as Duke of Milan) when the boatswain essentially disregards Duke Antonio because of his own knowledge of sailing. That the tempest occurs immediately in the play suggests that hierarchical disruption, misunderstanding, and chaos will be crucial to the play's plot development.
Chess
Chess was a common symbol in early modern English plays, and its role in The Tempest is no different. As a game that relies heavily on foresight and strategy, chess serves as a symbol of manipulation and control. This is precisely what Prospero sets in motion when he conjures the tempest at the beginning of the play; in the end, Prospero is revealed as the expert "chess player" because is able to win back his dukedom by manipulating and out-strategizing his opponent. Additioanlly, when Ferdinand and Miranda are seen playing a game of chess in Act 5, they speak innocently and vaguely about their romantic relationship. In this way, chess functions as a reflection of power romantic power dynamics, as the game revolves around being able to wield power without your opponent knowing.
Miranda's Virginity
Many of the male characters – especially Prospero – seem to be particularly interested in Miranda's status as a virgin. Her virginity is a symbol of her purity and innocence, but it is also an assurance that any children she has will rightly inherit the dukedom Prospero has worked so hard to win back. Furthermore, Miranda's virginity is also a symbol of her youth and passivity. By preserving her virginity, Prospero is able to keep her from becoming sexually autonomous and therefore maintains control over her body and her mind.
Water
Water is a central motif in the play. It is an emblem of mystery, wonder, transformation, and loss. When Ariel sings his famous "Full fathom five" song, for example, he describes to Ferdinand the transformation that has happened to his father's body on the sea floor. Because the play takes place on an island, water becomes a symbol of loss – drowned bodies are typically irrecoverable – that is then upended by the play's focus on recovery (Prospero's dukedom, the appearance of allegedly drowned characters, etc.).