Palmer Son 1 (Webisodes)

Pre-Apocalypse
Nothing is known about Palmer's kids before the apocalypse, but we can assume their father did the best he could to protect them.

After Apocalypse
Palmer's children were having a suprise party when the apocalypse began. Walkers attacked the party, pressumably killing the guests. The children were bitten and infected. The children were probably instantly turned and decided to attack their father. He probably locked them outside or in the cellar to prevent them from biting anyone or himself. They kill Palmer, and he asks Andrew to kill him.

Later when Andrew, Hannah, Jamie, and Billy are going to try and escape with Palmer's truck, Andrew goes into the basement of Palmer's house. Andrew is killed and eaten by the children. This is the last we see of them, but they are probably still roaming around the town, looking for fresh bait.