Every time a resident reaches the next level, a birthday cake will appear above their heads. Tap it to give them a birthday present. You will be given a choice between a new hat/hairstyle or new clothes. Occasionally, footwears or glasses will also be a choice. You will not be able to see the new look before making your choice. If you like it, tap "Awesome" and your resident will wear that from now until you change it on their next birthday. If you do not, tap "Keep the old one" and they will continue to wear what they were wearing before.
If you ignore the birthday bubble above their heads and leave the main area or close the game, you will usually miss your chance to give them a birthday present. If you wait to tap on the birthday bubble until they level up another level, you will only get one birthday bubble (for instance, if your resident reaches level 2 and you wait to tap the birthday bubble until they've gained 10 more xp reaching level 3, you will not get a second birthday bubble to celebrate level 3). Birthday cakes that you have in your inventory can be used to get the residents new clothing or hairstyles, but they will not level up if you do so.
The style of the residents living in collecter's item houses cannot be changed. They level up without birthdays. Birthday cakes MAY still work, but not all parts may be modifiable (e.g. In the chipmunk hut, the option to change headgear has no effect.