The advantage of hypertext is that in a hypertext document, if you want more information about a particular subject mentioned, you can usually "just click on it" to read further detail. In fact, documents can be and often are linked to other documents by completely different authors -- much like footnoting, but you can get the referenced document instantly!
To access the web, you run a browser program. The browser reads documents, and can fetch documents from other sources. Information providers set up hypermedia servers which browsers can get documents from.
The browsers can, in addition, access files by FTP, NNTP (the Internet news protocol), gopher and an ever-increasing range of other methods. On top of these, if the server has search capabilities, the browsers will permit searches of documents and databases.
The documents that the browsers display are hypertext documents. Hypertext is text with pointers to other text. The browsers let you deal with the pointers in a transparent way -- select the pointer, and you are presented with the text that is pointed to.
Hypermedia is a superset of hypertext -- it is any medium with pointers to other media. This means that browsers might not display a text file, but might display images or sound or animations.