Winter Bash is a fun, lighthearted event SE runs every year as a way of celebrating the many amazing people who make the communities awesome. It's open to any Stack Exchange site that chooses to participate.
During Winter Bash, users earn virtual hats by completing certain tasks. They're sort of like badges, but they're more visible and less permanent. Hats can be "worn" on users' avatars just about anywhere avatars show up on the network. Here's an example of what some hats looked like last year, taken from the network leaderboard:
Of course, SE is rolling out an all-new assortment of hats for this year's event, as they have in the past.
Individual users who don’t want to participate, don’t want to see hats, and/or are generally anti-hat will have an “I hate hats” option available. And just like last time, at the end of the event, all hats will go back into storage.
The only visual change to the Stack Exchange sites themselves will be the presence of the hats and the “I hate hats” button in the footer.