A furry is, at the base, a fan of anthropomorphic animal characters (frequently and somewhat ironically abbreviated as simply anthro). Exactly where on the Sliding Scale of Anthropomorphism these characters are doesn't matter, but most of them fall under Petting Zoo People and Funny Animals, and they're not on either extreme end. The characters may in some instances also be shapeshifters, transforming from fully human form to some kind of animal.
The majority of the fandom consists of people who enjoy fiction and art centered around anthropomorphic animal characters whether it be something as simple as text stories, art or comic books, or enjoying something as elaborate as furrydom's perhaps most significant de facto artistic and financial mainstream media success, the internationally popular Kung Fu Panda film franchise by DreamWorks Animation. They may simply take part in what others create, or they may be creators themselves. Many, but not all, also have a fursonanote short for "Furry Persona", which is a furry character representing the individual.
On the deeper end of the fandom, which is a minority, there are people with anthropomorphic animal fetishes, and a minority of them are erotic cosplayers. Despite the small size of this group, they have gathered the most attention in the fandom, much thanks toCSI: Crime Scene Investigation, ER, MTV, American Dad, Aqua Teen Hunger Force, Vanity Fair, Maxim, Something Awful, and numerous other programs. As such, furries are understandably distrustful of the media, and believe it necessary to refuse Convention access to known members of the press.
Another subculture of the fandom is Otherkin. These are people who believe they are, in real life, not actually fully human. It may be that they claim to have an animal spirit, which may or may not be a real animal (Therians limit themselves to real animals). Some go so far as to disown the human race entirely, and claim they've been born as the wrong species. Even Otherkin, who're mocked by other furries, mock these people, to give some context to where they are on the social ladder.
There is a significant stigma to identifying yourself as a furry, mainly because on The Internet, they are often the butt of many jokes. For this reason, there are a lot of people who have the same interests, but don't identify as furries. On the other hand, if someone does like works with furry characters, some people are inclined to label them a furry. They may like the works for the furry characters, but they may also like them for simply being great stories. Who exactly is a furry is up for debate, but the only people you can be certain of are the ones who identify themselves as such.
Also, meet Dr. Kathy Gerbasi, social psychologist to the furs.
Artistic License - Biology: Often used to point out common anatomical failures (rabbits with paw pads, male kangaroos with pouches) or more "human" traits (Non-Mammal Mammaries, feet on unugulates). Most people often justify it by saying "It's fiction" or "It's a toon" and pointing out that most anthros wouldn't exist in real life anyways.
Some reptiles tend to have smooth hair-less skin instead of scales, often justified by scales being too time-consuming to draw.
Bestiality Is Depraved: People who hate the fandom for its sexual side tend to cite this as their reason more often than not. Of course, the vast majority of the fandom prefers anthropomorphism to feral (four-legged, non-anthro) creatures, and generally considers bestiality to be depraved as well. Furries do NOT like it when animals are abused or victimized.
Cat Girl: The war-torn border of furrydom and often a very big target for haters, depending on the amount of cat features on the person (or the other way around).
Cowboy Bebop At His Computer: Almost every mention/article about furries in the media will suffer from this to some degree, even if the article is written by a furry (since not all furries are in agreement as to what the fandom is about). Making things more difficult is that furries have become very hesitant to appear on the news.
Fandom Rivalry: Putting furries and bronies in the same room is a bad idea, despite (sometimes because) of the overlap between the two fandoms. Many furries see bronies as being "in the closet" (as claimed by the flamewar-inducing Fur Affinity banner My Little Furry: Denial Is Magic), or get angry at furry image sites being flooded with pony pictures. Meanwhile, bronies tend to subscribe to the furry stereotypes and want to disassociate their own fandom with them.
Follow the Leader: Since many fursonas seem to start as glorified fan characters (very few will admit this), it may be possible to predict the trends of the fandom by the latest anthro-animal related media coming out.
Early sources for the modern fandom include Robert Crumb's Fritz The Cat and Disney's Robin Hood, both of which emerged in the early 1970s.
Furry Confusion: Often mocked. Some people run with this, their fursona is "feral" (as opposed to centauroid or anthropomorph), a normal-looking animal, just able to talk, and sometimes sports a hairstyle.
Guy on Guy Is Hot: In the fandom, furry yaoi is much more popular than furry yuri. This site shows that gay-themed furry comics are twice the number of hetero- and lesbian-themed furry comics combined.
Mix-and-Match Critters, Heinz Hybrid & Interspecies Romance: Furry "species" seem to be more analogous to human races than actual biological species, seeing as they're able to mate, interbreed, and even have viable children. One has to wonder why the whole "furry" race isn't composed entirely of hybrids.
I Am Not Shazam: If you get into technical aspects, "furries" refer to characters with fur (e.g., mammals). For reptilian characters, those are "scalies", as for the birds, "avians". The list goes on. However, a commonly accepted blanket term for a member of the fandom is "furry".
Image Boards: Fchan, Ychan, "The Goddamned Furry Board", Paws and many more not listed here. Some of these boards, not being monitored closely, host truly frightening and squicky art, so Google at your own risk.
Ink Suit Actor: Most furry artists depict "themselves" as furry characters, known as "fursonas", either due to the awkwardness of inserting human characters into a furry universe or just because they want to. Some are deliberately drawn to strongly resemble their creators.
Internet Backlash: Openly identifying as a furry is loaded. Although acceptance seems to be (very slowly) spreading, and even Something Awful tolerates open furries if they're civil, i.e. not an Attention Whore or complete nutcase, those are banned once they stop being entertaining. However, a good deal of the more civil sites seem to accept or ignore furries these days. It seems it's fairly safe to be a furry nowadays so long as you don't walk into the lion's den.
In other words, think of the internet as a workplace. Unless you're on a website devoted to that sort of thing, people don't want to hear about your fetishes or unconventional lifestyle.
Interspecies Romance: Very common due to the wide variety of species used as "fursonas" or in furry media.
Money, Dear Boy: Even if you don't really identify with the fandom, there's a pretty good amount of money to be made if you're a remotely talented artist willing to draw other people's fursonas. The more you're willing and able to contribute to draw naughty stuff, the more money to be made.
Non-Mammal Mammaries: Usually played to the ground, often a bit of a Base Breaker given how people point out that anatomically, it would be especially painful on avians (and by extension, gryphons) considering what mammaries are actually for...
Older Than They Think: The modern form of the fandom as we know it today was first established in the early 80s, but there were people who could easily be called "furry" in the 60s. However, if you consider the mere liking of something not human "furry", then you can go as far back as ancient times. The animal-headed Egyptians gods are the usual reference.
Our Centaurs Are Different: Often generalized as "taurs" to differentiate between the traditional exclusively-equine version. Usually have the lower body of a normal animal and the upper body of a human or more conventional furry. An example: "Equitaurs" have the lower body of a non-anthro horse, and the upper body of an anthro horse, differentiating them from the half-human Centaurs.
Power Perversion Potential: Many animals have uh... sexual traits, anatomy, and abilities that humans don't have. Some furries like to make anthropomorphic animals have sex. You do the math. Or don't.
Reclusive Artist: There are a few artists in the fandom who're sometimes notoriously hard to get a hold of because of all the drama that they've gotten into. These particular artists may be hesitant to interact with others.
Rule of Personification Conservation: Generally, a character or a cast of characters should only be animals if the plot requires it to (unless the author happens to be a furry, of course).
Rule 34: Although the fandom has shifted heavily away from being an actual fandom of existing works and towards being a community of creators who are also fans of each other, this is still plenty common.
Species Surname: Many furries will use a screen name that includes the species of their fursona, like "JohnnyCat" or "MartyMouse" or the like.
Stock Dinosaurs: Mostly because they're the most well-known ones. You'll see some lesser known ones every now and then but most commonly people pick Raptors because they're naturally bipedal.
Stop Being Stereotypical: For every furry acting outrageously and/or inappropriately, you can be sure that there are perhaps a dozen that Face Palm (Face Paw?) to the extent that a number of furries keep their 'furriness' a complete secret, for fear of comparison. What one furry does affects all the others. See Vocal Minority below.
Stripperiffic: Frequently, though far from universally. Sometimes justified though that some designs wouldn't allow for some clothing anyway. And after all, most furries would have to shave their bodies to actually show "skin". But hey, there's a fetish for that too.
Troll: To levels WAY past extreme. The trolling of the furry fandom on common web-sites has gone well beyond Acceptable Targets into rather horrific Memetic Mutation. Look at any video in YouTube featuring furries in any way, positive or not—the amount of trolling you get is simply baffling. Some of them are rather tame (by trolling standards), the others are rather extreme.
Uncanny Valley: This can happen to some people, although a lot of the more "toony" or stylized art doesn't tend to be that Uncanny. Realistic Art can be rather uncanny, but the most Uncanny art are photomorphs, especially because most people don't seem to realize that you can't just slap random animal features on humans to make 'em look like a furry and hope it looks decent.
Unusual Euphemism: All over the place, in both the traditional "awkward topics" sense and in more innocent things. Hands will be referred to as paws, feet will be footpaws, spouses will be mates, and some furries even prefer sillier euphemisms like "headfur" for hair. Usually done in a tongue-in-cheek manner.
Vocal Minority: The reason as to why the Furry Fandom has a bad image in the first place. "Normal" people aren't noticed as easily.
What Measure Is a Non-Cute?: The reason why there's so many foxes, wolves and cats in the first place. They're cute! It's also why birds, insects, spiders, Fish People, and whole lot of reptiles are rare in the first place.
Generically, some artists have actually left the fandom because their work started popping up on sites without their permission. There were several cases of people even selling prints of artists' work without their consent.
Some artists also leave the fandom because of the attitude of other furs, from mass idiocy to crass rudeness as reasons, dropping out of public and no longer showing up anywhere furry related. They will still keep furry friends, they just won't interact with the larger part of the fandom anymore. Other reasons could include not wanting to be associated with the fandom, for fear of being ridiculed or ostracized by people outside the fandom. The validity of this reasoning varies from situation to situation.
The Wiki Rule: WikiFur is basically the resource for information about furry cons and gatherings, although it usually trails off into stubs and vanity pages about anything else. Still, it's one of the few sites to start out hosted on Wikia and gain enough support*
read: Wikia drove them off with their advertisement barrage and domain takeovers