It's too bad that our library isn't currently up. There are a lot of words and I probably won't hit on all of them. This is only a brief list because there are a lot of words. I hope others jump in to clarify if I don't get these exactly right. I'd advise reading the other threads in Q&A if you haven't already. And ask questions if you run into something specific you want to know about.
Otherkin - a person who identifies as a nonhuman being, usually on a spiritual or psychological level, not physical. Otherkin are physically human. The identity, called a kintype, is not consciously chosen. Legitimate kintypes are an integral part of a person's being, as such they doesn't disappear over time. Otherkin usually, but not always, have certain experiences in common such as mental shifting, phantom limbs, a desire to return to some other "home" where they can live fully as their kintype, and occasionally display small habits or patterns pertaining to their kintype consciously or unconsciously.
Kintype - the species that an otherkin identifies as.
Therian - therians are similar to otherkin but their identity is related to animals that actually exist or have existed on earth. Someone who identifies as a wolf would be a therian, although they still fit under the otherkin umbrella. Someone who identifies as a dragon would not be a therian, but would still be otherkin. There's also a subset called "theriomythic" which might be used if the mythical animal has the mindset of a natural wild animal, for example a "wild" dragon as opposed to one who is intelligent and uses language.
Theriotype - the species that the therian identifies as.
Fictionkin and fictotype - identifying as a person, creature, or species found in modern fictional works. If the character in question is human then technically they don't qualify as otherkin.
Mental shift - a change in mindset closer to that of another species. These shifts range in intensity from barely noticeable to practically forgetting the human side, and everything in between. Extreme shifts are very rare. Usually a mental shift just involves something like a heightened awareness or quieting the human chatter in the mind. Sometimes a mental shift will let certain behaviors though, for example I might do a birdlike head bob or just seem to stare at nothing if I'm mentally shifted toward the bird side. Mental shifts can be encouraged to happen or sometimes they show up spontaneously. (Otherkin are still responsible for their own actions whether they are shifted or not. If they have trouble controlling their actions during a shift then they probably have other issues that they should speak to a therapist about.)
Phantom shift - a sensation in which a person feels that they have extra limbs that aren't really there. For example, many otherkin and therians sometimes get the sense of having a tail or wings or sharp teeth. The sensation ranges from having an odd undefined sensation in the general area, to having a detailed mental image of the limb and being able to move it at will. Some people talk about having pain when a "tail" gets stuck in a door or stepped on, but such vivid sensations are rare. Phantom limbs often just phase through other objects because they aren't physical. (Otherwise my phantom wings would make it difficult to sit in a normal chair.) Phantom limbs are something like a secondary mental image superimposed over the physical body.
Awakening - the point at which a person discovers that they are otherkin. Sometimes they know right away what their kintype is, and other times they have to put the pieces and clues together in order to narrow it down. It's not about acquiring anything new, just recognizing what has been there all along.
"Kin with" "Kinning" - the community has been around long enough for there to be a linguistic divide and this is where it is. These words are not considered legitimate by most older otherkin. Using kin as a verb implies that kintypes are acquired through some kind of action. That also implies that someone goes through a process of choosing their kintype. There's the admonition of "Don't kin with people outside of your race." (Which applies more to fictionkin than otherkin because otherkin are about species and not race to begin with.) The admonition is based on the faulty idea that otherkin choose their kintypes. This type of language runs the risk of reducing something that is an innate part of our being to being just a hobby or pass time of those who don't really understand it. These words seem to be used more often among fictionkin communities.