Romanticism in Music was an extension of the Classical era that it followed. Essentially, it was the increase in emotional content and use of "tone colours" and the decrease in strict form and structure. There was also an increase in nationalism in music.
Composers started to write orchestral music that was programatic, ie, music that tells a story, describes an idea or an emotion. To say that the Romantic Era and the Classical Era were completely different eras however is wrong. All of the same characteristics were there. Orchestration was similar, as was the make up of the orchestra itself, just the orchestras used by romantic composers was larger.
One main difference was the increased use of chromaticism in Romantic music and as the era progressed this shifted entirely from the Chord I, IV, V progression of the Baroque and Classical to the almost free form chord progessions of the late 19th century.
As for major representatives, Beethoven was one of the first and his nine symphonies beautifully show the transition from Classical to Romantic styles of writing. Chopin and Liszt for the romantic piano solos, Wagner for opera, Berlioz and Bartok for Orchestration. Tchaicovsky was in there for piano, romantic ballet, orchestration. Debussy and Ravel for impressionism and some of the late romantic composers such as Rachmaninov and Holst wrote some of the most beautiful music on earth (my opinion).
Hopefully that gives you some places to start looking further. Program music, nationalism and Impressionism were the 3 major movements of the Romantic era so in any assignment, I'd start looking there.