Trophies are a system on The Duck which rewards users for achieving various things. Generally the more active you are on the site, the more Trophies you'll earn. Trophies can be earned by uploading pages, achieving the higher comic ranks, posting in the forums, getting a featured comic, and more. Many of the trophies are handed out automatically as they are achieved.
The trophy system for the new site is still under development - so the earning of trophies is temporarily suspended. People who currently have trophies have carried them over from an older version of the site.