Inuyasha is the dog-eared half-demon title character in the anime/manga of the same name.
"犬夜叉" meaning, roughly, "dog demon".
Inuyasha appears essentially as a young adult human male, but has a few canine characteristics inherited from his dog-demon father. His ears are dog-like and sit on top of his head, and are inevitably fondled playfully by any modern-day humans he encounters. He has yellow eyes with vertical slit pupils. He has claws instead of fingernails, the better to scratch his enemies with. His canine teeth are pointier than a human's.
Our hero has long, white hair, like all the best anime demons. He has two distinctive snake-like locks of hair which fall down either side of his face, framing it. He almost exclusively wears baggy red trousers and a red shirt, with the Tessaiga sword at his belt. He also has a magic necklace which is impossible to remove, of which more later.
"Don't treat me like a dog!"
Inuyasha's red outfit is made of fire-rat fur, making it impervious to fire damage and much more resiliant in general than ordinary fabric. (If it weren't, it would be dissolved or turned to charcoal or generally torn to shreds every time Inuyasha fought a strong demon). The design of the outfit is very similar to the clothes worn by the priestesses of the period.
Inuyasha is armed with the Tessaiga sword, inherited from his father, and indeed forged from one of his father's fangs. The sword is magical and allows Inuyasha to perform various magical attacks.
The sword has a couple of anti-theft measures built in. Firstly, in its dormant state it appears to be an unremarkable, old, damaged sword. It only transforms into a GIGANTIC MAGICAL SWORD OF DOOM when Inuyasha is wielding it, and even then he has to be using it with good intentions or it will remain tatty and useless. Secondly, it cannot be picked up at all by one who is not at least part human (much to the disgust of Sesshomaru, Inuyasha's older, full-demon half-brother). Any full demons trying to take hold of it get a healthy dose of lightning.
The scabbard of the sword is also noteworthy; in order to contain the powerful Tessaiga and its anti-demon lightning curse, the scabbard itself is highly lightning resistant, and can be used as a kind of shield even in the absence of the sword.
In later episodes, the sword gains the power to destroy magical barriers. It glows red when it's gearing up to smash through such a barrier.
Inuyasha is a powerful half-demon; he has inhuman strength and agility, and recovers quickly from wounds. Barely an episode seems to go by without Inuyasha being horribly impaled, often multiple times, with various different implements, but no amount of impaling seems to slow him down significantly. He can leap incredibly high, but cannot actually fly. His doggy side gives him a powerful sense of smell which he can use for tracking.
The rules for using the magic time-travelling well seem to be complicated; for whatever reason, Inuyasha alone is able to go through the magic well to the modern era and back again at will.
"You guys really piss me off, you know that? You're always asking me 'can you sense this, can you smell that'!"
Naturally, Inuyasha spends a goodly portion of his time yelling the name of whatever crazy attack he's performing.
- Blades of blood! - Inuyasha can fling flying blades of his own blood from his claws, but he uses this rarely.
- Iron reaver, soul stealer! - This seems to be a fairly straightforward "claw the enemy with my clawed pointy claws" kind of thing.
- Wind scar! - This is a Tessaiga attack, where Inuyasha harnesses the power of the "wind scar," using his preternatural abilities to sense where the focal point of the division in the... anyway, he shoots devastating jets of flame from the sword, and it looks cool.
- Backlash wave! - Another Tessaiga attack. This time, merely by waving the sword around a bit, he turns the evil aura of his enemy against them and it turns into a big whooshy thing and smacks them around the head, or the tentacle, or whatever. This only works if the enemy is being evil enough to have an evil aura worth turning into a whooshy thing, though.
- Diamond Spear Blast! - This is an enhanced version of the Wind Scar, where the Tessaiga fires diamonds out as well as the blast, for increased damage.
Inuyasha wears a magical necklace, which is impossible to remove. When Kagome (and only Kagome) says "Sit!", then the necklace hurls Inuyasha to the ground, with some considerable force. The necklace was put on Inuyasha by the priestess Kaede to make sure Inuyasha played nicely with his new human ally.
One day a month, on the night of the new moon, Inuyasha becomes fully human. His hair turns black, and he has normal ears and eyes, and none of his demon strengths and abilities. He returns to his normal self at sunrise.
If Inuyasha becomes very agitated, and is separated from the Tessaiga, he can transform into a full demon. He gains red eyes, some eyeliner, pointier fangs, and incredible strength, but he loses what little brainpower he has and becomes a kind of slavering beast, just as likely to kill his friends as his enemies.
Inuyasha has a half-brother (same father, demon mother) named Sesshomaru. They don't get on.
Inuyasha is in love with the priestess Kikyo, or at least with the memory of her. Any fool can tell that he's also in love with Kagome, but he seems to go through spates of denying this.
Beyond his messed-up love life, Inuyasha has two main goals.
Firstly, he wants to take revenge on the demon Naraku, who has committed a variety of terrible acts, including one very personal to Inuyasha.
Secondly, he wants to assemble the remaining scattered shards of the magical jewel, the Shikon no Tama. He claims that he will use the reassembled jewel to turn into a full demon and shed his human side, but it seems more likely that his motivation is altruistic - he wants to prevent the shards getting into the hands of demons who will use them for evil. However, Inuyasha hates to admit to altruism, because it doesn't jive with the "bad-ass demon" image that he'd like to have.
Inuyasha is unarguably the most powerful member of his motley crew, and does the lion's share of busting demon heads. However, he's not really what you'd consider the brains of the bunch, and so doesn't really take the kind of role as a leader that you would expect. Instead, he seems happy to go with the group consensus for the most part, despite occasionally striking off on his own.
Born to a mighty general among dog demons, and a human mother, Inuyasha grew up something of an outcast amongst other "pureblood" demons. Inuyasha had a loving relationship with his parents but never gained the respect of his half-brother Sesshomaru, who despises him for being a half-breed. His parents now dead, he feels very alone, and finds it hard to make friends and trust other people.
Inuyasha is short-tempered, reckless, sometimes selfish, but essentially good, in spite of himself. Somehow the character is impossible to dislike, in spite (or perhaps because of) of his shortcomings. He claims he wants to get rid of his human side, and become a full demon, but he seems genuinely angry to see the strong preying on the weak, and will intercede to protect humans from demons. Part of his charm is his almost total lack of guile, although he does sometimes exhibit a certain base cunning when in a tricky situation.
Emotionally, he is clearly horribly torn between Kikyo and her reincarnation, Kagome, and this is perhaps the principal tragedy of Inuyasha. He can never have them both, and he ends up denying himself either of them.