I'd say the web actually started back in the mid to late 60's. In response to the USSR launching Sputnik, the U.S. government had formed the Advanced Research Projects Agency, (ARPA), for the advancement of science and technology for military usage. In 1969 ARPANET was commisioned by the Department Of Defense to conduct research into networking. In that same year three computers were linked together at UCLA, Stanford Research Institute, and University of Utah. The following is a description of what took place during that experiment.
What transpired there confirmed the idea and ability of computer networking. This first small step in my opinion, had broken ground for what we have today, although this network we now know as the world wide web is beyond anything that those scientists 40 years ago could have probably ever imagined.
Source:
"Roads and Crossroads of Internet History" by Gregory Gromov, Chapter #1
The thing that really made the idea of the World Wide Web workable was the developmment of a common language that could allow computers to communicate with another. This language, developed in 1974 by ARPA scientists, was know as transmission control protocol/internet protocol (TCP/IP). The idea behind TCP/IP was that:
This language contained the building blocks and framework for how information was to be distributed through the network, know now as the world wide web.
Source: Chapter Two: From ARPANET to World Wide Web Richard T. Griffiths (Leiden University)
As far as the person who contributed the most to the Web as we know it, I chose Sir Tim Berners-Lee. Among his greatest achievements, aside from his work with URls, HTTP and HTMl, I believe, was the founding of the World Wide Web Consortium, or W3,in 1994. Without bringing cohesion to the emerging internet browser companies, the web would have been left fragmented and exclusive. The W3 ensures the continuing consistency and stability of the internet as a whole. This is even more important now as the next phase of internet history is written. Some interesting facts relating to Sir Tim Berners-Lee:
Source: A Little History of the World Wide Web and BBC News