You may be looking for the Star Trek: TOS story Challenges.
Some challenges are gift exchanges, in which each participant produces a work as a gift for a specific fellow participant. Other challenges are prompt-based, in which each participant is given an assignment from a list created or compiled by the challenge mods. Challenges may also involve participants working together to create a larger work, something like a round robin. Remix challenges occupy an odd sort of niche, wherein each participant uses the work of another participant as a prompt, but the final product is not necessarily considered a gift.
History of Challenges
Fans have been "challenging" each other since the beginning of organized fanworks.
Fans in media zines sometimes wrote fiction in response to a single piece of fanart offered up by an artist or editor. Some early examples were in the mid-1970s in Interphase. These were sometimes referred to as Art Interpretations.
Exchanges or Ficathons
In an exchange of fannish works, frequently called a ficathon when fan fiction is exchanged, when participants sign up they provide both a request (specifying the nature of the fanwork they would like to receive: medium, genre, pairing(s) or characters, rating, and possibly a prompt) and details about the kind of fanwork they are willing to provide.
The mods assign a recipient to each participant, who must produce a fanwork meeting their recipient's request by a deadline. Usually the participants are required to work in secret, not revealing whose request they have received until after their assignment has been completed and the finished fanworks have been posted. If a fan is unable to complete her assignment, a pinch-hitter may be assigned.
The Secret Santa is a special kind of ficathon. The difference between a Secret Santa and a regular challenge is that the authors of the stories remain secret for some time after the challenge, and are then revealed, usually sometime around December 25. Occasionally this leads to controversy when it is pointed out that Secret Santa challenges may, through terminology or scheduling, inadvertently cause non-Christian fans to be or feel excluded.
Yuletide is currently the largest fannish Secret Santa challenge; the first online fannish Secret Santas were Secret Slasha (Buffyverse) and Don We Now Our Gay Apparel (popslash), both of which began in 2001. (if someone knows of an earlier one, feel free to edit).
Currently, most large fandoms have at least one Secret Santa; very large fandoms, like Harry Potter, may have several. There are also several multifandom Secret Santas like Yuletide and I Saw Three Ships.
The first minis community on LiveJournal was femslash_minis founded by Cadence K. It featured a series of short ficathons, with one or two week deadlines and low required wordcounts. The featured fandoms were the Buffyverse and Firefly.
A Big Bang is distinguished by its high required word count and the pairing of fan authors with fan artists. Authors usually have a preliminary deadline by which they must hand in a rough draft, which provides the artist assigned to them time to create an illustration or cover for their fanfiction.
Charity Drive Challenges
In charity drive challenges, participants list the types of fanwork they would be willing to provide, and other fans bid for the right to make a request. The winning bidder donates the amount of her bid to a selected charity. She then provides a detailed request to the fan who originally made the offer, and this fan eventually presents the winning bidder with a finished fanwork tailored to her request.
This type of challenge can be controversial because it contradicts the fannish principle that some types of fanworks, especially fanfiction and vids, should not be bought or sold. (The taboo against money changing hands when it comes to fanworks has historically not been so strong when it comes to fanart.)
The counter-argument is that since no fans are actually profiting from their fanwork in the case of a charity challenge, it technically isn't the case that fanwork is being "bought" or "sold." But the fact that money is involved at all (sometimes in extremely large sums) still causes some fans to worry that these types of challenges may draw the attention of TPTB and open up the participants to legal risk.
In a fest, the mods solicit prompts, collate them, and provide the list to participants. Each prompt may be claimed only a limited number of times (usually once or twice, or once each for fanart and fanfic). Participants may be assigned scattered due dates, resulting in weeks or months of daily fic, or all fic may be due on one day.
pick a prompt (any prompt, even if it's your own) and write the porniest bit of fiction you can, or make the hottest manip or painting or vid or song. Make it as kinky or as subtle as you like, but make it hot, melt your readers, create a stampede to all the fannish bunks worldwide.
Other fans periodically run similar challenges, not all of them adult-oriented.
Theme challenges are ones in which participants make a claim, such as a character or pairing, and create fanworks about that claim while inspired by a set of various themes or prompts. The first themed challenge community was 30 Kisses, which was founded in early 2005. It inspired the creation of many similar challenges such as 31 Days and Icon Fiend 100.
The first prompt table was Fanfic 100, billed as "the ultimate FanFic challenge." Like theme challenges, participants claim a theme, e.g., a character, pairing, or fandom, and must complete a set number of fanworks to various prompts. For instance, a Fanfic 100 participant who claimed the pairing Spike/Buffy from Buffy the Vampire Slayer would have to write 100 Spike/Buffy stories to such prompts as heart, moon, and months.
Drabble and Flashfic Communities
There are many Icontests (Icon Contests) on LiveJournal. The mod provides a weekly prompt, and members create icons that meet that prompt. Members then have a chance to vote for the best icons.
LIMS stands for Last Icon Maker Standing. These are ongoing icontests; icon-makers sign up at the beginning of a cycle, and each week the icon maker whose icon(s) garners the fewest votes is eliminated.
A bingo challenge is one in which themed prompts are distributed as bingo cards. The bingo cards are then filled out by the creation of various fanworks to accrue points. The first bingo challenge was Kink Bingo.
Many websites or communities may have a list of prompts that passers-by or members are welcome to fulfill at any time. For instance, Yuletide invites writers to fulfill New Year's Resolutions year-round by writing unfulfilled prompts from the previous year's challenge.
Some people feel that due to so many challenges, there isn't enough spontaneous fic writing happening in fandom.