Toy Day begins at 6PM on December 24. On that day, you have to dress up in the Santa Claus outfit and give gifts to your villagers. Afterward, Jingle the reindeer will give you prizes.
Before Toy Day, there are a couple of things that you should do to get ready.
Starting on December 1, your villagers will tell you what they want for Toy Day. Be sure to make a screenshot or a note of their requests. You will need to remember the list on Toy Day!
In order to participate in the gift-giving on Toy Day, you have to have the full Santa Claus outfit. The Able Sisters start selling the parts of the Santa Claus outfit in December. There are five parts to the Santa Claus outfit:
Starting at 6PM on December 24, you will find Jingle the reindeer walking around near your town tree. Talk to him to start participating in Toy Day.
When you talk to Jingle the first time, he will mention dressing up as Santa. Put on your Santa outfit and talk to Jingle again. If you are wearing the full Santa outfit, Jingle will give you a gift bag. It contains all of the gifts that your villagers asked for.
Now go around and give your villagers the gifts that they asked for earlier in December. Be sure to give the gifts before 6AM. If you give somebody the wrong gift, the villager will still accept the gift, and you'll still get a reward from Jingle, but you won't get the special bonus prize.
After you are done giving the gifts, go talk to Jingle. He will give you either some Festive Furniture or some Jingle Furniture.
If you gave all of your villagers the correct gifts, you will get a picture of Jingle in the mail the next day.