"Romantic music" is, alongside "Classical Period" and "Baroque" music, one of the three big, traditional groupings of classical (as opposed to Classical Period--the terminology confuses people) music. Baroque is most well-known for Bach, Vivaldi, Telemann and co.; the classical period, roughly corresponding to the Enlightenment, is represented by Mozart, Haydn, (Quantz and C.P.E. Bach), and arguably Beethoven. Romantic music is the broadest and least well-defined of the three terms; it is traditionally typified by Wagner, Beethoven, Brahms, and Berlioz, the last three of whom formed the original "three B's". It was perhaps the first movement in music in which the composers consciously tried to express the same ideas as their contemporaries in other arts, literature, and philosophy, and it thus provides a much better comparison to literary romanticism than, say, classical music does for enlightenment philosophy.
Early romantic music can be defined with similar characteristics to the ones we used to define literary romanticism: it is often nostalgic and pastoral, it aims for the "sublime and beautiful" (which Dostoyevsky mocks), and it tries to evoke emotions more than previous movements in music.
To give a few examples of how all of this manifests in the music of the period, I think that it's a good idea to look at the transitional figures to see how the changes in their styles were essentially Romantic. The quintessential transition figure to Romanticism is Beethoven, who initially imitated Mozart and his idol Haydn but later developed a very individual style that precipitated romanticism. His first few works, including his first four symphonies, were largely classical in style; he stuck to traditional forms, and his works were consequently more or less formal and traditional. However, in his "middle period" from 1803 to 1814, he began to transition to a more romantic style. In each of his symphonies, he tried to break some tradition: in his fifth symphony, he pioneered the use of a motif (probably the most famous motif ever) to unify the piece; in his sixth, he wrote the first explicitly programmatic symphony, that is, the first that attempted to conjure a specific image; in his seventh, he uses dance rhythms in all of his melodies; in his ninth, he invented the choral symphony. Arguably, each also represents a further development of romantic music; the sixth and seventh share a similarly nostalgic, pastoral tone, and the ninth is an evocation of the sublime.
Here is a recording of his Symphony no. 6 "Pastoral":
The work is fairly lighthearted for Beethoven. It portrays a journey through the countryside, starting with the composer's feelings in the first movement ("Awakening of cheerful feelings upon arrival in the country", a name which, on a side note, was amusingly modified to "Awakening of cheerful feelings upon the arrival of the Finnish spring" to disguise performances of Sibelius's Finlandia under Russian censorship), and progressing through a "happy gathering of country folk", a brook, a storm, and a "shepherd's song", all almost comically sentimental portrayals of rural life that are very characteristic of the early Romantic period.
Just to confuse anybody who has actually read this far, romantic composers kept doing their own thing long after nobody wanted to associate with romanticism in literature or philosophy; the musical versions of movements like realism (in music, verismo), nationalism, and even modernism are all grouped under "romanticism" because it came to mean a theoretical and formal style rather than a subject matter; works composed in modern times with romantic-period instruments and tonal patterns are still called "romantic" even if they have nothing to do with literary romantic concepts--musical romanticism sprung partly from its literary counterparts, but they didn't follow one another closely for long.