Bollywood is the nickname given to the Indian film industry - it's a play on the word Hollywood. The B comes from Bombay (also known as Mumbai), a big city in India.
Bollywood is the informal name given to the popular Mumbai-based Hindi language film industry in India. The name is a conflation of Bombay. The old name of Mumbai, and Hollywood, the center of the United States film industry. Though some purists deplore the name (arguing that it makes the industry look like a poor cousin to Hollywood), it seems likely to persist and now has its own entry in the Oxford English Dictionary. Moreover, Time Magazine recently cited that the Indian movie industry produces over 1000 movies every year—nearly four to five times the output of Hollywood—making it the largest film industry in the world.