I see now. It's not that Internet Explorer isn't working or that you're being kicked offline but that you're having DNS issues. The WWW isn't the Internet in itself but is only a small part of it all. That you're able to send and receive email is an indication that you're still on the Internet. You should probably contact your ISP and get them to help you determine why your DNS is responding so strangely.
Just as a point of reference, if you're on a high speed Inernet connection, such as cable or DSL, as long as your computer is turned on and the lights on your modem are lit, you're very likely always going to be on the Internet. You won't have to be running Internet Explorer (or any other browser) and viewing Web pages to be on the Internet though you would need to be on the Internet to view the pages. Consider that you can have legs but not run while you can't run without legs. Clear as mud? Good stuff.
Call your ISP. They're likely the only ones who can help you with this problem.