Text web browsers only display text on web pages, ignoring all multimedia such as flash, images, video, etc.
Why would you use a text browser when there are graphical browsers? If you are a webmaster you would be interested in knowing how your page would look to Google and Yahoo! crawlers. This would help you in SEO (search engine optimization). If you are a researcher, you need access to journals and research papers. You can access them through your university or research institute as they have subscription but you cannot access these papers from you home or neighborhood coffee shop. To access these papers, you can log into your server using you terminal and use a text browser to access these papers.
In short, text browsers are not for common users.
Following is a brief list of text web browsers:
Edbrowse
Perl-based with an editor, browser and email client.
Elinks
Tabbed Text browser
Emacs/W3
Ideal for Emacs users
Links
Text and graphic browser with a pull-down menu system.
Lynx
Free text only web browser and Gopher client distributed by the Internet Software Consortium. Probably the best available.
w3m
Text-based web browser and a test formatting tool.
WebbIE
Web browser for visually-impaired and blind.