Online forums that exist expressly for the purpose of posting images and/or discussions. They usually have a variety of sub-boards, each with its own focus. Unlike most forums, threads are never archived by the site, so if nobody posts in one for a while, it will eventually be deleted.
The imageboard as it's usually thought of today began in Japan. Futaba Channel, a spin-off of the popular text-based discussion board 2-Channelnote Officially pronounced "Ni-Channel", but most English netizens are comfortable with just calling it "Two-Channel" (2ch), is probably the best-known. Sometimes the blanket term 'wakaba style' board is used (wakaba referring to the board software that runs the board), traditionally allowing anonymous posting with a self-enforcing theme and standard that hopefully keeps it from becoming too mediocre.
The best-known in the US is probably 4chan, created in 2003 to serve as an English-language version of Futaba Channel. Especially among Americanotakunote Or, although the term is used more in a negative sense, "Weeaboo", it's quite popular for the generation of memes and, due to anonymous users and loose moderation, extremely variable quality; for these reasons, one could see imageboards to be the antithesis of social networking sites such as Twitter and Facebook.
Most American wakaba boards trace their heritage to 4chan, as alternate boards proliferated when 4chan went offline in its formative years. The successful ones tend to cover more specific niches, like Meganekko or sad girls in snow. Or guro hentai. Or lolicon. Or guro hentai with underage meganekkos in snow. Or guro hentai with underage meganekkos in snow weeping profusely over image boards. Or... you know what, let's just stop talking about it.
The only real standing rules of just about every imageboard site in existence (besides specific topic rules for subforums) is that its users can't post anything considered illegal in a site's area of hosting (many admins turn a blind eye to things like unauthorized filesharing, however). Dodgy to outright highly illegal material is occasionally posted anyway just for the pleasure of subverting this rule; mods unsurprisingly counter this with anything from warnings to reports to the FBI.
What many imageboard sites have in common is using the same distinctive layout and interface from 2ch: there is a sidebar either on the left, bottom, or top (but never the right) that lists each board on the site in a simple URL theme (usually a contraction of their topic name), and to the right of this is the actual site content. Most boards are also set up fairly similarly to 2ch (/v/, /m/, /tg/, etc.), so you can usually count on wholly different sites to have the same general setup.
Not to be confused with our own Image Links Wiki, or Image Booru.
Notable Imageboards (every link below should be considered NSFW unless specifically mentioned otherwise):
While not an imageboard itself, The Overchan V.2 used to maintain an (as up-to-date as humanly possible) list of every English-language imageboard in existence (including furry imageboards and text-based 2ch-esque boards). It was also the closest thing to a worksafe link from this point forward. Two replacements fortunately have popped up: AllChans and Overchan v3
The aforementioned Futaba Channel, a spinoff of the text-only, and far more infamous, 2ch. Take note that unless you live in Japan, you can't post at either board, though you're free to lurk. This is to prevent crapflooding and DDoSing from foreigners outside of Japan. And even if you could, it is much more chaotic, vitriolic,somehow more organized than 4chan, and farless tolerant of newbies than 4chan is - for a board already disdainful of newbies. Most of the imageboards frequently have discussions and images (or ASCII art) that are unrelated to its intended purpose and spam is basically omnipresent. It has a lot more boards than 4chan and churns out memes, trolling, racism (to the point where there is an entire subboard on 2ch/an dedicated to bashing Koreans), and .gifs at an even faster rate. Unlike its western counterpart, it still maintains its guro boards to this day.
The also-aforementioned 4chan, which hosts boards dedicated to just about every generalized subject you can think of (and a few specialized subjects, as well), is home to the self-proclaimed "final boss of the Internet" (the Random [Bullshit] board, aka /b/), and is the starting point for about half the memes in the various subpages of the Memetic Mutation article. Humorously enough, 4chan was created by a bunch of regulars from Something Awful's anime subforum, a fact that is best left unmentioned on both sites. Has its infamous, luckless spin-off, 7chan, Which has its own spin-off, 99chan (see below). The users consider them 'better' than 7chan - and sometimes even 4chan.
420chan, created as a result of one of 4chan's longer downtimes, is a site specifically geared towards drug users (with several boards dedicated to different types of controlled substances; the site refuses to allow people to directly buy or sell drugs through the site itself, though). It also has several non-drug-related boards, including the site's most second-most-visited board, /wooo/. /wooo/ is one of the most well-known Professional Wrestling imageboards on the 'Net, to the point where WrestleCrap considers the board their Arch-Enemy and forbids their Message Board users to mention it. (420chan's most visited board is, as you might expect, /weed/.)
fchan, which is the most well-known furry (primarily erotic furry) imageboard out there; although assumed to be created as a response to 4chan's infamous "April Fur's Day" pranknote The notoriously anti-furry 4chan created a furry board, /fur/, on April Fools' Day...but then kept it up for a few more days. Then on April 4th everyone who'd posted on /fur/ was banned from the entirety of 4chan., it was actually in existence for a while prior to it (and was created as a result of 5chan — which hosted a couple of furry-based boards — dying). The site set something of a standard for future furry imageboards by segregating content into different boards (one for female-only art, one for male-only art, and so on); it also marked the beginning of the term "DNP" ("Do Not Post") in the Furry Fandom, due to the list of artists who asked that their work not be posted on the site. fchan's heavyhanded moderation and large DNP list led to several furry imageboards coming into existence to counteract fchan's perceived stupidity, the most notable of which are:
Lulz.net, which is sometimes referred to just as /furi/ after the site's main board; it is pretty much always EXTREMELYNSFW, and not always because of furry porn — and not always because of porn. Many of the board's threads are about gore pictures, grossout images, trolling, and extremely vitriolic politics. There's also more furry-hate on the site than /b/ and Something Awful combined. Even the board's regulars will often admit that the site is basically "/b/, but with less furry porn". If you go in without Adblock prepared beforehand, it's also a great stress test for your antivirus application of choice.
55chan, a Brazilian imageboard. Before the site died, it used to attract a good number of functionally illiterate users, commonly assumed coming from the social networking site Orkut. Those users would usually flood the boards with bland topics and requesting the other users to raid their enemies' pages on Orkut. The moderation eventually put a stop to this by enabling a filter that asked the user to copy a simple sentence using capitals and punctuation without using the copy-paste command; only once this had happened would a user be allowed to use the boards. That filter was enough to fend off the vast majority of the problem users. Its replacement is now BRchan.
7chan, an imitator of 4chan and purportedly luckless when it comes to actually being recognised. Usually spread by annoyed ex-/b/-tards fed up with the decline of 4chan. Boasts most of the content 4chan does as far as subboards are concerned. Due to lower visitor numbers tends to be less chaotic. Really doesn't like Something Awful or the notion it's a 4chan snowclone. Maintains a Lolicon board (among other things) but will ban anything that even smells funny of CP.
888chan was the imageboard chiefly used by Project Chanology, an Internet-based movement based on the Anonymous "culture" borne from 4chan's /b/ (the group has no defined leaders, and most of the members of the various Chanology subgroups do their best to remain anonymous) which was (and still is) dedicated to "fighting" the Church of Scientology. (Project Chanology itself has been going for more than two years as of this writing; it has used several different imageboards as their base, with 4chan's /b/ being the original starting point of the movement.) As of 2013, 888chan now has a board for lolcow discussion and harbors the remnants of 789chan's users. Was shut down on March 31, 2014.
Desuchan, the result of a certainmeme becoming too big and obnoxious for the other imageboards to stand. ~desu (pretty much worksafe) It also has its fair share of spinoffs, mostly as backups during downtimes.
Krautchan, pretty much THE German imageboard. Became famous in the wake of the Winneden school shooting, when the media fell for an extremely obvious fake post announcing it (a real but completely unrelated post photoshopped). Has an /int/ board. Its humour consists mostly of intentionally bad and literal translations of english memes and imageboard slang. (Beweis mich falsch, fappieren)
Another British Image Board is B3TA, which in addition to message boards, also has weekly or so Image Challenges, quizzes, movies and games. While the vast majority of the userbase are okay folks and the image challenges bring out funny images.
plus4chan, originally started as an offshoot of 4chan prime's /co/ (Comics and Cartoons) board, has since grown into its own standalone site with a very tight-knit userbase. Much of the content is still /co/ related, but there are also boards about anime, video games, cooking, music and television, and Homestuck, among others.
99chan. Elitist as hell, and damn proud of it. 99chan is even slower than 420chan, but has a huge variety of boards. Expect to be banned if you post demotivators, memes, or if the many mods are feeling bored. Most of the humor on the site revolves around inside jokes and self-deprecation. Practically everything is decided on the IRC (better known as the "circlejerk"), which is self-admittedly populated entirely with pseudo-intellectual gay Canadian alcoholics. Tends to have a "sekrit club" mentality, so if your post smells even faintly of 4chan, expect to be banned.
MikuChan is an image board dedicated to Hatsune Miku and other VOCALOID characters/voice banks
789chan, an imageboard with a focus on technology. It is, however, most well-known for its /cwc/ section, which, of course, focuses on Christian Chandler of Sonichu fame. Despite the title, very few threads are about Chris, and generally discusses several other people on the internet that users don't like. Was shut down on November 2012.
Nido is a Chilean imageboard. Even with the NSFW material that pops frequently, it's pretty lighthearted compared with most of the imageboards it tries to emulate. With a permanent hatred of all the people that commit spelling errors (they have an explicit rule against it, and if you post with any spelling errors you will get banned), it is the most active imageboard from that country to date. It was made by some users unhappy with 6-chan (another Chilean imageboard) when that had legal issues. Most of the boards are locked for people that don't have a Chilean IP address, but they have an /intl/ (international) board that is open for outsiders.
Shrekchan a strange and somewhat active board with a focus on all things Shrek. Mostly full of shrek-themed images and NSFW greentext stories with a... unique version of chanspeak. Yes, it's ironic... Probably.