General Fan Fiction Archives
The West Wing Fan Fiction Archive: The largest storehouse for TWW fic on the net. We're talking lost-in-the-stacks large.
Jenny's West Wing Fan Fic Archive: Another big fiction archive.
The Best and the Brightest: As the name suggests, a very selective archive for The West Wing fanfic.
Our Boys: Selective archive of fiction for The West Wing and Sports Night - some slash, some not, all excellent.
TWW Collective: A list of all the West Wing mailing lists.

Shipper Archives
Greens and Blues: TWW shipper fanfic - all couples.
Sam Loves Mallory Land: Sam and Mallory site.
Bipartisan Relations: Sam and Ainsley fiction site.
My Funny Valentine: A CJ/Sam fanfic site.
The Sam Seaborn and Donnatella Moss fanfic archive: Sam and Donna fic site.

Slash Fic Archives
If you're new to fanfic and don't know what slash means, it can easily be described as: Sam and Josh (or Sam and Toby, or Jed and Leo, or Abbey and CJ, you get the point) in love. And oftentimes, they demonstrate it explicitly. So, if you're under seventeen, or if you're offended by this idea, do not click on these links.
Missing Angel: Main archive for West Wing slash.
Writer's Block: Anna-Marie Jennings's site (Sports Night stories too)
Jae Gecko's Fan Fiction
lowercase k: k's really more of a Sports Night writer, but she's an amazingly good Sports Night writer, and she's written a couple Sam/Josh stories which shouldn't be missed.
snk: all the imaginings: S.N. Kastle's stories (including Sports Night and X-Files)

Other Character Archives
Claudia-Jean.net: Aside from the obvious, this site also hosts the MSEB and JSEB, and if you don't know what those are, you're not properly in fandom.
Oddsmakers: CJ/Toby fiction archive.
Hot When You're Like This: Another CJ/Toby fiction archive.
Josh Lyman Fan Fiction Archive: Go here for more Josh/Donna links, because I'm not a shipper. That said:
Jo March and Ryo Sen
LeoMcGarry.net: Leo fanfic and much, much more.
Both Sides Now: Jed/Abbey fiction archive.

GENERAL SITES FOR THE WEST WING

Official Sites
NBC's West Wing Site
Warner Bros. West Wing Site

Unofficial Sites
All of these sites are fantastic resources of information for fanfic writers (episode guides, transcripts, who's who, which assistant goes with which staffer, job descriptions, guest stars, a layout of the West Wing... You have a question, there's an answer here somewhere), as well as up-to-date news. As opposed to us - we have two-day-old news, usually found on one of these sites.
Continuity Guide for The West Wing
Inside the Bartlet White House
JoshLyman.com
The West Wing Online Message Boards
A la Maison Blanche: A French WW site.

Missed an Episode?
Television Without Pity: Very detailed, very funny recaps of the show, plus message boards where Aaron Sorkin himself comes out to play.
epguides.com: Episode guides - titles, airdates, cast and crew lists - for more shows than I thought existed.