In most cases it is not realistic to run a web site from your own computer, even with a broadband connection. That's why companies still offer a 'hosting' service, where you rent space and facilities on their computers (rather like renting a unit in an office block).
Authoring
Building a web site is a cross between desktop publishing and programming. The degree of control over the layout of a page is more limited than with desktop publishing. And things are made more complicated by the need to produce sites that are compatible with many different web browsers, and also adhere to guidelines for visually impaired users. One distinct advantage of publishing on the Web is that each page is a separate document, and so can usually be modified independently of other pages (rather like swapping pages in and out of a loose-leaf binder). Where the Web scores highly is with the almost limitless scope for adding interactive functionality, e.g. questionnaires and user-maintained contact databases.