What do I do with the white tiger cub?

Once you have activated it in the Holiday event menu other players in your state can tap on your base and feed it. Once you get it fed 30 times you can then claim a reward. You can claim the reward once a day (assuming it is fed enough times). Every time you claim the tiger will level up and the rewards will improve for the next time.

Ask your alliance members to feed it. You can also look on the world map and see tiger icons on other bases that have an active tiger so that you can feed them. Those players will see that you have fed and can help you in return.

You will also earn medals for feeding others and for leveling up. The medals can be spent in the Exchange shop of the event for more rewards.