The first movie was made in 2013 and it was not called Bollywood at that time. There was no visible industry as such in those days , Its only after 1950 the industry really took off and known or referred by people as Hindi movie industry.
Probably between 1970 and 1980 some journalist or movie critic tossed the name Bollywood. Nobody is sure when and who exactly named Indian Hindi film industry as Bollywood. To answer who first made a movie in India is Dada Saheb Falke who made the first silent movie in 1913.
Bollywood is not like Hollywood. Bollywood is not even a place. Its just a name for all Indian movies and music. Although Mumbai in India is the movie making capital of the world. But Bollywood is indeed not a place!