I'd say the web actually started in 1984 when Tim Berners-Lee considered the problems of information presentation: physicists from around the world needed to share data with no common machines and no common presentation software. He wrote a proposal in March 1989 for "a large hypertext database with typed links." His boss, Mike Sendall, encouraged him to implement his system on a newly acquired NeXT workstation. After considering several names, he settled on World Wide Web.
The thing that really made the World Wide Web workable was the onset of companies such as Compuserve, America Online, and Prodigy, which added millions of new users to the web in 1995, quickly making it a household word. These subscription services made it possible for people to connect across the globe.
The person who contributed most to the Web as we know it was Tim Berners-Lee because he developed the first web browser on a NeXT computer which included an integrated editor that could create hypertext documents. This lead to the development of the Mosaic browser, which in turn lead to the creation of the Netscape browser, Internet Explorer, and eventually Firefox.
The concept of a home-based global information system goes at least as far back as Isaac Asimov's short story "Anniversary" (Amazing Stories, March 1959), in which the characters look up information on a home computer called a "Multivac outlet," which was connected by a "planet wide network of circuits" to a mile-long "super-computer" somewhere in the bowels of the Earth.