I'd say the web actually started in 1957 in the Soviet Union with the launching of the Sputnick satellite. This was the earliest stage of the internet, which can be defined as a network of interconnected networks that share information through Internet Protocal. The original term for the internet was "ARPANET," which stands for Advanced Research Projects Agency Network.
The thing that really made the World Wide Web workable was dial-up telephone lines. In 1962, J.C.R Licklier was the first to develope a network of computers capable of of using ARAPNET. The commincications through this network were put to use in December of 1969 when four computers in major universities went online using the dial-up method of connectivty.
The person who contributed most the the Web as we know it was J.C.R Licklider of MIT because he was the first to connect public users to the internet.