Your birthday is a special event in Animal Crossing: New Leaf. You set your birthday when you create your character and have that character talk to Isabelle. You can receive special items on your birthday. This page has the details.
On the Saturday before your birthday, visit K.K. Slider at the usual time (8PM to Midnight) in Club LOL. On this day, instead of taking requests, K.K. Slider will perform "Your Birthday". He won't give you a copy of the song that day, but instead will send the song to you in a letter the next day.
A villager with whom you have medium to high friendship will be in front of your house when you start up the game on your birthday. This villager will take you to his or her house for your birthday party.
During the party, check on the birthday cake on the table. You can type in a wish, then stand in front of your birthday cake and blow into the 3DS microphone to blow out the candles. Then the villager hosting your birthday party will give you a gift! Depending on your friendship level with the villager who is throwing the party, you may receive one of these items:
The host of the party might also ask you what you wished for. It's up to you to tell the villager your wish or not!