Symbolism, Imagery, Allegory
Prospero's books are a pretty big deal in this play. They're the source of Prospero's magic, which is why Caliban says Prospero is completely vulnerable without them:
Why, as I told thee, 'tis a custom with him,
I' th' afternoon to sleep: there thou mayst brain him,
Having first seized his books, or with a log
Batter his skull, or paunch him with a stake,
Or cut his wezand with thy knife. Remember
First to possess his books; for without them
He's but a sot, as I am, nor hath not
One spirit to command: they all do hate him
As rootedly as I. Burn but his books. (3.2.11)
In other words, without the contents of his treasured library, Prospero's a "sot" (a stupid fool) and as powerless as Caliban. When Prospero says he's going to retire from the magic business, he promises "I'll drown my book" (5.1.5).
As useful as these books are, we also want to point out that Prospero's nose was buried in these very same texts back in Milan when his brother was busy stealing his dukedom (1.2). By his own admission, when Prospero was "rapt in secret studies," he neglected his duties as a ruler and isolated himself from the rest of the world (1.2.10). That's odd. Why would Shakespeare warn us that burying oneself in his or her art can be dangerously isolating? Let us know when you work that one out for us.
Brain Snack: The artsy 1991 film Prospero's Books concentrates on the (imagined) contents of our favorite magician's library.