A web slice is content on a web page which a user can subscribe to. The content is then available from a button in the Internet Explorer 8 Favorite's toolbar. When the content is updated, the button glows orange to alert the user that there is new content. When the user clicks the button, they see a drop down window with the updated content of the web slice.
They show how to use PHP to create the content to fill a slice with some simple content. A slice isn't much more than a part of your web page with a div tag classed as "hslice". IE8 can then pick that part of the site up and save it to the user's browser. You can see en example here.