An acronym meaning Amateur Press Association or Amateur Press Alliance, depending on who you ask. This is a form of self-published magazine with a built-in discussion forum. A given number of people join, and send in whatever they have regarding the focus of the APA to a member called the Central Mailer (CM). The CM collects all of the submissions, which are either copied by the Central Mailer (if the CM works at Kinkos, this is easy) or each person copies it themselves to a given copy count. The CM then gets it all bound, and mails everyone a complete issue, made up of their individual submissions. The Central Mailer also usually handles a treasurer function in managing everyone paying for their submission and mailing costs.
In an APA, you will often find introductions (in a member's first submission), fiction, essays, personal stuff, artwork, and other stuff. While most APAs that I personally have heard of relate to roleplaying, it seems reasonable to believe that many other topics would support them. The APA FAQ claims that there are APAs relating to politics, sexuality, and even religion.
What makes an APA different is mailing comments. When people create their submission, they also comment on other people's submissions from the previous issue, ranging from an in-depth discussion to the often dreaded RAEBNC.
Apparently, APAs came about during the 19th century through hobbyist printers who wanted to save postage
while distributing samples of their work. The first sci-fi/fantasy APA was formed in 1937, and was named the Fantasy Amateur Press Association.
APAs usually have a set limit on the number of members, producing a waitlist for those who have an interest when the member list is full. Members have to maintain a minimum level of activity, or minac, to not be kicked out, though members may quit at any time.
As an example: a new APA is forming, Tarn Rider, devoted to Gor. After finding ten people (the CM decides to limit it to
ten members) willing to admit that they actually enjoy the Gor novels, he sets some rules (like how often issues will come
out, etc.). Each of the ten people send in their material, the CM shuffles the papers and then mails one to each member.
According to the APA FAQ referenced above, the best way to join an APA is to get a copy of the New Moon Directory or join the APATALK mailing list.
Update: I have noticed that with the continued rise of the internet that APAs, and indeed the printed fanzine in general, seem to be in decline. Whether or not this is a good thing is debatable; there are more people posting bits and pieces they have created, but there are fewer connections between them.
Example taken directly from the All of the Above, the GURPS APA, web page at http://www.io.com/~rgilson/AOTA.htm (disclaimer: I am a member of All of the Above)
For more information, check out Amateur Press Alliances at http://www.spies.com/~rawdon/sf/apas.html, the APA FAQ at http://www.spies.com/~rawdon/sf/apainfo, the New Moon Directory at http://members.aol.com/NewMoonDir/, or the sizeable list of APAs from Gothik at http://members.tripod.com/~gothikapa/apalinks.htm.