Design: A completely new website, with everything made "from scratch" (images, content, etc.)
Redesign: A "renovation" of an existing site without much change to the existing content. This will include new navigation and/or images, fresh color scheme, etc. but keeping the same content (text, basic "feel" of the site) intact.
There is sometimes a fine line between new Designs and Redesigns, contact us and we'll help you determine which category your project would fall under.