This is from memory so corrections welcome! The internet more than just websites, its used to describe a whole plethora of 'online' services such as email, bulletin boards etc.
In the 1960s universities in the USA developed a system of sharing information between computers via phone lines. This was obviously a closed network, but a network nonetheless.
When Tim Berners-Lee worked at CERN in Switzerland he developed the scripting language HTML, this enabled websites to be created and shared across multiple platforms (windows, mac, UNIX etc) as it was designed to be generic. This is what really kicked off the internet as we know it. I think this was back in 1992.