Otherkin are people who, in some way, do not define themselves as human. A popular form this takes is that one believes one's soul to be that of another creature; equally popular is the belief that that one has had past lives as that creature. In the general case, Otherkin hold their identity as a spiritual belief. Note, however, that all generalizations are false; I personally know of several Otherkin (most of them dragons, but see also 2 The Ranting Gryphon) who are also atheists.
"Otherkin" is, above all, an identity. An Otherkin, at some level, by definition considers xirself to be other than human; "human" is not an accurate or adequate adjective. This is generally expressed as from some form of intuition, although some Otherkin initially decided that "human" was inadequate after failing to fit in with society in general.
Otherkin of almost any species have been observed. Dragons and elves are probably the most common, although gryphons and faeries are also seen in many places. (Please note that this could be an observation bias, as the author of this writeup is zirself a dragon and therefore tends to hang out in places where dragons congregate, so is more likely to see them.) There is significant debate as to where therianthropes fit into the category; the closest thing to a consensus I've been able to figure out is that therianthrope is a strict subset of Otherkin. (A full discussion of therianthropes should remain in that node; in short, one who identifies as a were-animal for some animal that exists on Earth.)
Most Otherkin are quite philosophical, as might be expected. They also tend to be rather bookish, and a disproportionate number find themselves at the head of their classes in school or college, frequently being placed in classes above their levels. A penchant for science fiction and fantasy novels is common, although video games are also prevalent.
The large number of online fora centered around Otherkin has given rise to a complete subculture. Notable features of it include a very strong priority on creativity and self-expression; glamourbombing is a popular activity. Politics within such communities can get nastier than one might expect at first; if I may be a bit bigoted here, I'd like to observe that it tends to be worst among elves. The community attempts to be welcoming and open to all comers, but this has the unfortunate side effect that people who ask too many questions about the beliefs of newcomers- especially if any doubt is publically expressed- tend to meet disdain, even when those questions have definite merit. (I, personally, was kicked out of a LiveJournal community for siding with the werewolf observing inconsistencies in the stated past of a werehobbit new to the community.) Discussion is generally encouraged, but participants are frequently expected to never tell anybody they're wrong and instead attempt to somehow express conflicting opinions in ways that do not contradict others'. This tendency has been decreasing recently, however, and skepticism is beginning to get slightly more welcomed.
Otherkin are quite frequently accused of being insane, deluded, crazy, or fucking nuts. These accusations are only rarely made by actual psychologists; many Otherkin who have informed such of their beliefs have been informed that there's nothing wrong with them. An inhuman identity is not itself a mental illness. This is not to say that all Otherkin are sane; I personally am acquainted with one presently institutionalized for schizophrenia. The critical thing to note is that the mental illness is largely disjoint from the Otherkin identity; even without it, xe would have been diagnosed as schizophrenic anyway. However, many Otherkin struggle with depression. An interesting correlation is an extremely disproportionate number of dragons, but not other varieties of Otherkin, being autistic or having Asperger's syndrome.
There is no great Otherkin conspiracy to take over the world from you puny humans. Move along. There's nothing to see here. I said move it. Stop reading this paragraph. This sentence was deleted for security reasons. Fnord.
Humor aside, a common misconception about Otherkin is that they all hate humans. While it's certainly true that some are misanthropic in this way, the vast majority are not. "Human-bashing" sentiments are not usually taken well within Otherkin communities. It is fair to say, however, that the vast majority of Otherkin consider humans to be something different from themselves. (Few, however, attempt to contest that they are physically human.)
It is certainly reasonable to debate whether or not Otherkin are, indeed, inhuman at some level. Many Otherkin are far from certain. This is an issue for the spiritual, the religious, and the philosophical to decide; it is not within the domain of science. "My soul is not human" is not a testable hypothesis and therefore has no assignable truth value by the scientific method. It cannot be evaluated by science; this does not, however, make it inherently invalid.
What cannot be denied is that Otherkin, in general, tend to be different from most other people. We're a weird bunch. Strange thought processes, an inability to just blend in to society, unusual fetishes, weird senses of humor, and bizarre moral codes are quite common. Many Otherkin who "come out" to their friends or family hear the quote
"I'd never believe it from anybody else. But you..."
because they tend to be exactly the person you'd expect to not quite be human. Very few Otherkin attempt to disguise their identity, to people other than their parents, in any way other than not telling people about it; most will answer honestly if asked directly.
Considering that, "Are they really inhuman?" is, at best, a moot point. Whether or not their spiritual belifs are objectively accurate is irrelevant when they do not behave or think in ways that a normal human would. "Human" is not a particularly useful label for people who only fulfill it in physical appearance. There is no pressing reason for such people to consider themselves to be human, and it is a poor choice to expect them to react the way a normal human would.
An interesting conclusion is my personal experience in my Psychology class. Dr. L. did a number of in-class experiments as a first-hand demonstration of people, including you personally (for "you" defined as "the people in the class"), act in typical manners. I was a significantly-outlying point on every single one...