Prometheus Unbound is a romantic poem in the modern sense of "love and romance" and in relation to the Romantic movement, which had more to do with whimsey and creative passion.
This second conception of Romance is the simplest to demonstrate. Shelley was a member of the second generation of British Romantic poets, along with his friends John Keats and Lord George Gordon Byron, who followed the first generation of Romantic poets, including Willam Blake, Samuel Taylor Coleridge, and William Wordsworth.
The Romantic movement in poetry came about in large part as a reaction to the Industrial Revolution and the emphasis on cold efficiency that followed. This shows clearly in the themes and references used by writers, especially in the second generation— Shelley being the most extreme of the three in most ways. This appeal to creative passion and imagination is clear in Prometheus Unbound from the first stanza:
Monarch of Gods and Daemons, and all spirits But One, who throng those bright and rolling worlds Which Though and I alone of living things Behold with sleepless eyes!
The more other meaning of romance comes into play with Shelley's description in the poem of the "Spirit of Love." One of many such symbolic allusions in the poem, Love is held as paramount—characterized as being more powerful than hate or aggression and boosted as the most important thing in all creation.
The easiest way to define Percy Bysshe Shelley's poem Prometheus Unbound as a Romantic poem is to look at the poet himself.
P. B. Shelley was a Romantic poet. His works were those typical of the movement based upon the fact that he wrote with language filled with references to nature. Shelley was simply an idealist (to the point of being heavily criticized--most notably by Mark Twain).
As for the poem, Prometheus Unbound, one can easily see Shelley's attention to detail regarding the natural elements surrounding Prometheus. Prometheus was bound to a cliff, by Jupitor's order, given Jupitor's jealousy of his power. For thousands of years, the Titan was beaten by nature itself (his heart being eaten, repeatedly, by an eagle and the elements of nature itself).
Overall, Shelley's intent for the poem was to show the power which individualism and intuition has on a person. Both of these themes were very important to the Romantic.