The main rec.puzzles.crosswords competitions are the ACC, CWC, PCWC, PVPCWC, SCWC and TCWC.
In an ACC, the moderator has considerable freedom in imposing a particular task on the setters. The winner of each round is invited to set and judge the next round.
In a CWC, the moderator sets two words: one is intended to be easy to clue and the other more awkward. The best single clue wins the contest, though a winner for each word is announced.
In a PCWC (People's Clue Writing Competition), unlike an ACC or CWC, submitted clues are made publicly available before the winner is selected. Readers of the newsgroup (except the moderator) then vote for their favourite clues. The winner sets the next word, but judging is always done by the people.
A PVPCWC (The People Versus The Professionals Crossword Writing Competition) is run on the same lines as a PCWC, except that the moderator enters a clue which a professional crossword setter has written.
An SCWC (Simple Clue Writing Competition) is like a CWC, except that entries are posted to the newsgroup, and the timescale is tighter.
A TCWC (Themed Clue Writing Competition) is like a CWC, except that the moderator sets a theme, which the clue should use.
If you are to moderate one of these contests, ensure that you don't set any of the answers that have been set before.
In an FSCC, the moderator supplies an incomplete crossword, and an entrant must complete the grid and write all the clues.
CDC (Cryptic Definition Competition) was started in response to a desire for clues of the "cryptic definition" (CD) type, a type not supported by canonical American cryptic crosswords. The only clues allowed are CDs.
NACC (Not American Clue-Writing Competition) was in part an experimental effort to attempt to see more clues outside of the American canon.
Karen Lingel ran KKK (Karen's Kryptic Kontest) where each entrant had to enter clues for a proper name of his choice.
There have also been collaborations designed to construct an entire crossword by team effort.
In a CCCCC (Collaborative Cryptic Crossword Constructors Championship), or C5, the idea is to construct an actual 15x15 cryptic crossword puzzle through the collaborative efforts of all the participants over a period of several months.
In a TCCCC (Team Cryptic Crossword Construction Competition), three or more clue-writers work together to create crossword puzzles modelled on those in various publications. The judges are professional puzzle editors and constructors.
UCPP (Usenet Crossword Publishing Project), was a collaboration designed to create a crossword good enough to be published.