While the letter B is crammed with composers whose music interests me, I find myself surprisingly underwhelmed by most of the composers in the C's. Maybe your interests will lead you in different directions, however. So I give you...
Cage (John), an American, twentieth-century, experimental composer who toyed with the very definition of music. Some of his work for "prepared piano" (a grand piano with a number of items placed on its strings) is actually quite interesting to listen to. His later works focused on the element of random chance in music (aleatoric music), and included pieces in which performers had to interpret a pattern of lines on a page; a piece for 12 radios that depends on what happens to be playing on certain frequencies at certain times; and the famous 4'33" - in which the performer(s) doesn't/don't make any sound at all. Some would say this was more theatre than music; or perhaps it was an object lesson in Cage's Zen philosophy.
Carter (Elliott) is an American composer. His early influences included Ives, Stravinsky, and Hindemith, but he has developed an individual style of atonal, serial music (generating entire pieces of music out of one chord or chord progression). Wiki says his music is lyrical, contrapuntal, and rhythmically complex, and I take their word for it.
Castelnuovo-Tedesco (Mario) was an Italian-born composer, descended from Spanish Jews, who settled in America after the rise of Fascism in Italy. He wrote operas, concertos for piano, violin, one and two guitars, and music for some 200 Hollywood films.
Chabrier (Emmanuel) was a French Romantic composer. He is famous for writing stupid operas with beautiful music. I can't think of a single piece by him that is especially well-known, but he influenced other composers, and his music is still recorded and played today.
Though Charpentier (Gustave) lived into the 1950's, he is best known today for his opera Louise, which appeared in 1900, and the popular soprano aria "Depuis le jour" from the same. He devoted most of his energy to teaching and didn't compose very much. I mainly mention him lest you confuse him with
Charpentier (Marc-Antoine), a Baroque composer of mainly vocal music (operas, sacred music, secular cantatas, etc.), notably the opera Médée. One of his themes is used as the jingle for the European Broadcast Union.
Chausson (Ernest) continues our parade of easily-confused, second-rate French composers. He was a late-Romantic type who wrote mainly songs, choral works, and chamber music. He did complete one symphony before his untimely death in 1899.
Cherubini (Luigi Maria) was Beethoven's favorite composer. An Italian-born musician who had a predominantly French career, Cherubini majored in opera and church music. What little of his music is still performed includes one symphony, a Requiem in C minor, and another opera titled Médée.
Copland (Aaron) wrote some of the most widely-recognized music in the 20th-century American idiom. He wrote symphonies, concertos, ballets, operas, and tone poems, including Appalachian Spring, Rodeo, Billy the Kid, El Salón México, and Fanfare for the Common Man.
Corelli (Arcangelo) was a Baroque composer who is mainly remembered for his sonatas for one and two violins and his 12 concerti grossi (a musical form contrasting a larger group of musicians against a smaller group). His surviving music is still played and recorded, and is quite beautiful. But Corelli is more significant for the influence he had on the development of Baroque music and violin technique. Music theory students also remember him by the characteristic cadence known as the "Corelli clash."
Couperin (François) is often called "Couperin le Grand" in distinction from his less-accomplished relatives. He wrote a book on harpsichord technique that revolutionized keyboard playing by introducing the use of the thumb! He also wrote four massive books of harpsichord music, plus some chamber, organ, and sacred music. He influenced many later composers, from Bach and Brahms to R. Strauss and Ravel.
Cowell (Henry) was a progressive American composer in the twentieth century. He also wrote an important study of the music of Ives and of world music. Cowell's imprisonment on morals charges from 1936 to 1942 marks a watershed in his career. Before this his music was radically experimental; afterward it was somewhat more conservative, though he continued to be a pioneer in combining Eastern and Western musical styles. He wrote 15 symphonies, among many other things.
Crumb (George) is another experimental American composer, still living, whose works include Black Angels (written for an "electric string quartet"), Ancient Voices of Children (accompanied, in part, by a toy piano), and four books of pieces for "string piano" (that is, a grand piano played as a string instrument rather than a keyboard) titled Makrokosmos.