The Quick Reply box normally appears underneath topics allowing a member to post their reply on the same page as the actual topic, rather than having to select the reply button that leads to the posting page to add a reply.
Quick reply can be disabled, enabled, and enabled but collapsed by default for any members by applying their preference for it in their profile.
Administrators can also set the default display of quick replies by a per-theme basis (Admin > Themes and Layout > Member Options).