The list is a community wiki; feel free to edit it if you have more information about a hat. If you like to do some research, most hats have been featured in earlier editions, so you might be able to find more information in old Meta Stack Exchange questions.
let's keep them in alphabetical order, for easier search
- "And YOU Get A Hat!" is a reference to an Oprah Winfrey meme based on an episode of her show where everyone in the audience was given a free car. The hat is designed like Oprah's hair.
- "Explorer" is a Pith Helmet, often worn by various jungle explorers. Explorers often visit less-trod ground, where it's unlikely to meet other people (who can upvote your answer).
- "Hero of Time" is a reference to this Zelda character.
- "Fascinator" is a reference to the hat worn by Princess Beatrice at the royal wedding in 2011. A "fascinator" is a term for this type of hat that is largely decorative.
- "Fashionable" is a reference to Iris Apfel as a fashion icon.
- "Foot of the Rainbow" refers to one of the myths about the rainbow, that a pot of gold can be found at its end. The hat is obtained after getting a gold badge.
- "IDENTIFICATION DIVISION" is a reference to Admiral Grace Hopper and COBOL.
- "It's-a me!" is meant to resemble famous video game plumber, Mario's, hat and mustache. The name is a reference to one of his catchphrases.
- "Like Clockwork": a reference to A Clockwork Orange. The hat is worn by the film's protagonist.
- "Milliner": someone who makes women's hats.
- "Think!" is a song by Aretha Franklin (the hat's original name, in 2013, was R-E-S-P-E-C-T, another song of hers).
- "This is fine" is a meme and an easter egg on the Stack Exchange chat server.
- "Waffles" are a long running meme on the Stack Exchange network. And they are yummy.
"Mother of Dragons" is a reference to the Game of Thrones character Daenerys Targaryen who has three dragon children.
The picture of "Ooh, Shiny!" is the moderator hat, given to ♦ moderators, whether they were appointed or elected, are active or retired. One of the things ♦ moderators do is closing and deleting posts. "Shiny" refers to the diamond.
"Red Hat With White Fur Trim" is a literal description of the Christmas hat; you got the hat by 'participating' on Christmas day.
The "Rep Cap" is an abbreviation of 'reputation cap', the informal name of the daily reputation limit.
"Row Of Many Candles" shows a Menorah, one of the items associated with Hanukkah; you got the hat by 'participating' on the last day of Hanukkah.
The "Some Kind Of Sun Thingy" shows a sun on it. You get that hat by participating on any site on December 21, the Solstice. This is the day with shortest period of daylight in Northern Hemisphere and longest period of daylight in Southern Hemisphere.