Simply put an Internet Cafe is a place where people can go to use the Internet. This is not a "virtual cafe" or any kind of an on-line thing. It is a physical place where people can physically go and use the net.
The term comes from the fact that traditionally (if something so new can have a tradition) Internet Cafes have been in actual cafes. Places that serve coffee and other things that go with it.
But an Internet Cafe could be a restaurant, a bar, even a bowling alley. The key is that it is a place where people can go and use the Internet in a social amosphere.
Internet Cafes serve a number of purposes. First it is a place where people can socialize and enjoy the atmosphere while they surf the net. Second it provides Internet access for those who do not have it elsewhere. Frequently Internet Cafes have training and help available for the new user.
This second purpose is an important one. When the World Wide Web came into existence it became possible for people who were not technically trained to access and use the resources available on the net. But still they had to be techincal enough to get access to the net in the first place.
Usually this meant investing in a computer system and dealing with the technical issues of getting connected and on line. The Internet Cafe bridges the final gap between the public at large and the Internet. In a way the Internet Cafe serves the same purpose as the FAX machine at the local copy shop. It provides access to the technology without the individual having to invest in or fully understand it.