From Nookipedia, the Animal Crossing wiki
In Animal Crossing
Unlike the player, villagers do not celebrate their birthdays. The player is sent a large, inedible cake from their mother, which can be displayed in their house. The player will also receive letters from villagers with presents and be greeted at the player's front door by another villager who will personally present a gift, often an NES game.
In Wild World
This is the first game in which villagers also have birthdays - the player can visit them in their house on that day, where they will be walking around their home with a fellow villager that they get along with, usually one of the same gender. The player can give any item to the villager that is celebrating their birthday for a gift, but this may cause the game to freeze. The two villagers in the room, if near to each other, may applaud and laugh. A player can tell if it is a villager's birthday by a letter sent by them in advance, and a notice that appears on the bulletin board on the day.
The player themselves receive a birthday cake which cannot be eaten, when they start the game on the day that is assigned as their birthday. The player will also receive a letter and a present from their mother.
In City Folk
Unlike in Wild World, villagers do not celebrate their birthdays, despite a notice on the bulletin board that congratulates them. Instead, a player can send the villager a letter on that day. The villager then thanks the player for wishing them a happy birthday in their reply.
As with Wild World, the player is presented with a cake on their birthday by a villager, which can be placed in the home or eaten. After eating, flowers briefly blossom from the player, much the same as the emotion "delight."
In New Leaf
When it's the player's birthday, a villager that has high friendship with the player will be waiting outside. The villager will then take the player to his or her house for a birthday party. During the party, if the player checks the birthday cake on the table, the player will be able to make a wish and blow on the microphone to blow out the candles. The villager that hosts the party will give the player a gift depending on how much friendship they have. The player may receive one of the following:
The host may also ask the player what he or she wished for, but it's up to the player if he or she wants to tell the host.
If the player visits K.K. Slider the Saturday before the birthday, K.K. Slider will sing "K.K. Birthday". He won't give the player a copy, but the following day, the player will receive it in the mail. 
In Amiibo Festival
If the month during which the game is set contains the birthday of one of the amiibo figurines, special event spaces marked by a cake will appear on the game board. Landing on one of these spaces will award the player Happy Points — with the player whose amiibo's birthday is that month receiving the most points. In addition, villagers on the game board have random events for their birthdays.