Goldfish were a species of fish native to Earth; they were renowned for their short memory. The First Doctor stated he wasn't a goldfish when it was suggested he had forgotten the events of a mere moment ago. (TV: The Romans)
The Doctor stormed into the TARDIS with a bowl containing a goldfish when Amy answered the phone. He pushed the bowl into Amy's hands and snatched the phone from her. He assured the Prince that his mother was safe aboard the TARDIS. Shortly thereafter, he realised he had the wrong fish. (TV: Night and the Doctor: Bad Night)
Behind the scenes
According to the Torchwood website (which is not considered a valid source on this wiki), while working for Torchwood Three, Suzie Costello began buying goldfish in bulk and killing them to test out the resurrection gauntlet.