I would say the web actually started in August 1962 with memos written by J.C.R. Licklider of MIT describing his "Galactic Network" concept. He described a vision of social interactions across a global network. As the head of computer research at DARPA, Licklider went about convincing others of the importance of his networking concept.
The breaktthrough that really made the World Wide Web workable was the packet switching theory. Leonard Kleinrock of MIT published his first paper on this theory in July 1961. Kleinrock theorized the use of packets instead of circuits for communications. The need for packet switching was confirmed in 1965 after an experiment of connecting a TX2 computer in Massachutes to a Q32 computer in California. The computers "talked" to each other over a low speed dial-up telephone line, which proved that the circuit switched system was inadequate for the exchange of information and programs.
The person who contributed most the the Web as we know it was actually not a person, but the Advanced Research Projects Agency (ARPA) which is part of the U.S. Department of Defense. ARPA funded the way for implementing ARPANET which became the Internet that we know today.