The Tempest is actually classified in Shakespeare's first folio as a comedy, which would be fine enough, except this play has certain elements that are peculiar to a new genre. When The Tempest came out, the "tragicomedy" had recently been brought into the English theater scene (by John Fletcher, who would eventually replace Shakespeare as principle writer for the King's Men). Its principle elements were pastoral settings (shepherds, shepherdesses, etc.), misunderstandings or mix-ups about love, and potentially tragic consequences that are happily avoided by some magical intervention. Shakespeare, because he's just like that, added to the form.
The Tempest is also part of a group of four plays (including Cymbeline, The Winter's Tale, and Pericles) that literary critics refer to as the "romances." (Not the kind of romances that feature a scantily clad woman and guy with bulging muscles on the book cover.) These plays were written at the end of Shakespeare's career and share a few things in common. Let's take a quick peek at our handy-dandy checklist of elements that are common in Shakespeare's "romance" plays to see how The Tempest fits into the genre: