The age old "zinger"
The 3 second goldfish memory myth was made popular in recent years when scientists on the "Mythbusters" television show. The term "memory like a goldfish" has been in history; somewhat of an insult or "zinger" to the person on the receiving end of the comment (due to the historic belief their memory spanned a grand total of 3 seconds).
The Mythbusters crew started a craze when they debunked the theory by teaching the goldfish to swim through an underwater maze. Their findings clearly showed that goldfish memory is actually quite longer than 3 seconds, as the fish were able to improve the time they could complete the maze each consecutive time they went through.
Before the myth was publicly debunked another test was performed: Back in 2003 a Plymouth University study was done that showed a goldfish was able to be trained to bump an underwater lever to release food. The goldfish was started out with a lever placed in their bowl at a set time of day, to the point where the goldfish was found to anticipate the lever being placed in the tank. After the goldfish memory was conditioned: The lever could be placed in the water at any time and the goldfish recognized it as a way to get food.
What does all this mean?
To most people it's just a silly, meaningless myth. To goldfish owners that think they have just another dumb, soul-less fish in their tank it might mean your goldfish has the potential to be taught a few tricks, or that you've been right all along about your belief that your goldfish can recognize you and maybe enjoys seeing you through the tank.
Debunking the goldfish memory myth also inspired tests on other fish, particularly in wildlife fisheries. Believing fish have a memory and can be trained can mean easier harvesting for testing or food processing.