A capella literally means "in the style of the chapel," for that is where this musical genre started -- in the choruses of the cathedrals and churches of Europe. Eventually the term came to stand for any person or group who was making music with only one instrument -- their voices. Some a capella singers also accompany themselves by using their voices, and other body parts, to create a musical range which they will use to underscore (or play under) their main vocal performance. When you sing a capella, your voice does not emulate a particular instrument but rather creates a musical line of its own.
Over time a capella singing began to be associated with college glee clubs and local civic organization singing groups. Barbershop quartets practice a specific style of a capella singing. In the 1950s, with the rise of street corner "Doo-Wop" music thanks to groups like The Platters, Frankie Lymon and the Teenagers, and the Del-Vikings, a capella music took on a new (and slightly tougher) persona. These a capella groups were known for their tight harmonies, markedly different from other similarly themed styles like scat and rap.
One of the most famous a capella musicians is Bobby McFerrin. After rocketing to success in 1988 with "Don't Worry, Be Happy," McFerrin would go on to a long and accomplished career picking up 10 Grammy awards and introducing people around the world to this unique musical art form. McFerrin often improvises in performance, uses all parts of his voice, and may tap on his chest in accompaniment.