Feel free to edit this to add more hats!
hit the daily reputation cap
Earn 200 reputation on a single site, within one day (as defined by UTC Time).
ask and answer your own question, each with a score of 1 or more
Do the Swim!
review 5 items
Complete 5 or more reviews of any type.
earn the tumbleweed badge
Earn the "Tumbleweed" badge, which states:
[Ask] a question with no votes, no answers, no comments, and low views for a week.
earn 20 hats
From Michael Hampton's answer:
You need to earn 20 hats, after the start of Winter Bash, on a single site. You then earn The Milliner hat on that site. (Which means you could earn it more than once...)
This is the same way the Johnny Three-hats hat works.
steal an accepted answer checkmark by posting a better answer
Post an answer on a question that already has an accepted answer, and have the poster of the question accept your answer over the previously accepted answer.
participate in closing or deleting a question
Adapted from How do I earn the "Passed Judgement" and "Reanimated" hats?
[You can earn this hat from] flags (as pointed out by PeterJ) and close-votes, and the question does actually have to be closed for you to earn the hat. The same applies to "Reanimated" and reopen votes...deleting your own question does not give you the hat.
participate in reopening or undeleting a question
Adapted (heavily) from How do I earn the "Passed Judgement" and "Reanimated" hats?
Vote to undelete or reopen a question. The question does actually have to be reopened or undeleted for you to earn the hat. Undeleting or reopening your own question does not give you the hat.
have a user account created in the first month of the site's existence
According to balpha'a answer on How do I earn the Old Hat hat?:
the user's creation date (hover over the "member for" value on the use profile for the precise time) is before the time when the site's Community user (user id -1) turned 31 days.
Anna Lear says:
...private beta sites launched during Winterbash will be excluded from the event and you can't earn any hats on them.
All known secret hats:
How Do I Get the Winterbash 2013 Secret Hats?