A static website is an organized collection of simple files, much like documents on your own computer, made available on the Internet. When a someone visits a page on a static site, they are simply downloading and viewing an HTML file and usually a few other associated files such as images and stylesheets, just like you would download any other file. This is the extent of the technology, and its simplicity is its greatest strength.
This is in contrast to a dynamic website, which is far more complex and relys on a database and scripting language to deliver its goods. When a visitor visits a page on a dynamic website, files are also downloaded, but these files are often assembled on the fly by the scripting language querying the database. Interaction with a database makes many things possible: shopping, account management, and social interaction to name a few. But dynamic sites are subject to hacking, slowness, and instability, and must be vigilantly maintained.
Static sites are by no means replacements for dynamic sites. They are simply more appropriate for applications where extensive visitor interaction is not required. For such sites, databases and scripting languages introduce unncessary technical complexity, and in turn unnecessary burdens on site owners and visitors.