The festival takes place on the beach between 10am and 6pm on the 18th day of Spring. On the 17th, Popuri will come to your farmhouse in the morning to ask what duck you'd like to enter. If you own a duck and want to enter the festival, choose a duck and Popuri will take it for the festival on the following day. Popuri will not take ducklings or sick ducks. If you do not own a duck, Popuri will still encourage you to come watch the festival.
On the 18th, go to the beach and talk to Popuri to start the judging of the contest. Your duck will be up against three other contestants. The higher the heart level your duck has, the better your chance of winning. A duck at 9-10 hearts will almost be guaranteed to win (although 9 hearts may still get you second place). Any placing besides first place counts as a loss. Nothing will happen if you lose.
If your duck places first, it's level will increase and you will gain 100 Farm Degree points.