There are dozens of websites where you can read full books online. Project Gutenberg http://www.gutenberg.org/ is probably the largest, but there are many more including Arthur's Classic Novels, Classic Bookshelf, East of the Web (short stories), the Baen Free Library (science fiction) and Page by Page Books.
You can find a reasonably extensive list of these sites with links at: http://www.nku.edu/~gregoryj/lit/txtsrc.shtml
With the exception of the Baen Free Library, the sites above contain mostly older works that have fallen into the public domain. For newer copyrighted books, try your local library. Many libraries are now using a service called OverDrive to provide their patrons with downloadable ebooks, audiobooks, music and video. Many also have ebooks through NetLibrary. If your library has either service, you can use them from home for free.
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