some giving tree that almost every ward in America does at Christmas time.

Step 1: Put the money in an envelope and tell the Bishop you want it to go to so-and-so
Step 2: Bishop goes to their house and says "someone wanted me to give this to you anonymously"
Step 3: Bishop says a prayer of thanks for kindhearted people

It could alternatively be any person who you trust to do it anonymously. The Bishop is just an added level of anonymity, and someone that many people trust to not steal the money.