Basically there's 2 collection quests, one for the snowman and one for the X'mas tree.
For the snowman making in Physis, there are four phases:
Phase 1- 300 snowballs, Phase 2- 200 snowballs, Phase 3- 100 snowballs, Phase 4- 30 X'mas Decorations.
When you put the materials needed to the unfinished snowman, you will obtain the keyword: "Christmas"
For the X'mas tree making in Emain Macha, there are only 2 phases:
Phase 1- 50 X'mas Decorations, Phase 2- X'mas 100 Decorations
When you put the materials needed to the unfinished X'mas tree, you will obtain the keyword:"Decorations"
How To Make Snowballs: During the event, Physis region will have snow you can collect with empty hands, then you use handicraft to make it into a snowball.
How To Make X'mas Decorations: Collect 1 Thick Thread, 5 Branches, 5 Nails, and 5 Small Gems then use the handicraft skill to make it.
Use the keywords when talked to the Santa's Helper and you will obtain your rewards:
Christmas: Short Reindeer Hairband (Copper) or Long Reindeer Hairband (Gold).
Decorations: Reindeer Hairband (Silver) and the title "Merry Christmas".
*The completed X'mas Tree and Snowman will return back to its original state every 6 minutes in real life.
*One character can only use the keyword to receive the reward once.
And the new pet will be added during the event will be the Reindeer and the Christmas Bear(Already released last time).
(What are you thinking about? >_>)