Every time I think I've done justice to my "general introduction to fine-art music," I realize a couple days later that I skipped some minor, or even major, composers who just have to be mentioned. Here are a few more of them, but just a few whose names begin with the letter A!
Adams (John) is a still-living "minimalist" composer. That means his music generates a variety of effects out of a minimum of thematic material. Actually it's hard to pigeon-hole Adams; he doesn't exactly fit the minimalist mold. But let's just say his music can be very exciting. His operas include Nixon in China, The Death of Klinghoffer (about the Achille Lauro tragedy), and Doctor Atomo. Some of his other works are Shaker Loops, Harmonielehre, The Chairman Dances, and Short Ride in a Fast Machine. I myself sang in a performance of his cantata On the Transmigration of Souls (written in memory of the victims of 9/11).
Albéniz (Isaac) was a Spanish pianist whose works evoke Spanish folk music. He wrote many piano pieces (e.g. Suite Española and Chants d’Espagne) and even some orchestral works (e.g. Spanish Rhapsody and Catalonia), but his best-known work is a set of twelve piano pieces titled Iberia.
Albinoni (Tomaso) is most famous for his "Adagio," which is frequently heard in various instrumental combinations, but that piece is not really by him; it was written by another composer named Remo Giazotto, though allegedly based on fragments of Albinoni's music found in the ruins of the Dresden library after the Allied firebombing in World War II. Don't worry, though; Albinoni wrote a good deal of other music, mainly in the style of Baroque concertos, which is widely played and recorded today.
Alfvén (Hugo) was a Swedish composer, watercolor painter, and acclaimed author whose five symphonies and numerous picturesque "tone poems" are worth hearing. His Swedish Rhapsody No. 1 is quite popular, and so is the guitar solo by Chet Atkins based on it. Remind me to do a post on "pop music borrowings from fine-art music" sometime.
Arnold (Malcolm) wrote a lot of lighter art music, toward the "popular" end of the spectrum. For example, he composed the scores for such films as The Inn of the Sixth Happiness and The Bridge on the River Kwai (for which he won an Oscar). He also wrote some popular dances and pieces youth orchestras like to play. On the more serious side, he is best represented by his nine symphonies. He also wrote concertos for clarinet, guitar, and harmonica!