Claymores
Clare is a half-human, half-monster warrior known as a Claymore. Created by the Organization, Claymores are all female; they're women who were taken in as little girls and transformed into half-breeds, by having the flesh and blood of yoma implanted inside them. Claymores almost never become what they are willingly--most often they're orphans who are taken in by the Organization, or girls who are sold to the Organization by their families or villages. Clare is the only known exception to this rule, being someone who approached the Organization and asked to be made into a Claymore, so that she could hunt the Awakened Being Priscilla.
Claymores look a little different from humans. Their most notable features are their silver eyes and pale hair, which leads them to be referred to as "silver-eyed slayers" or "silver-eyed witches". They also have some kind of stigma or scar on their abdomen that is usually covered by their uniform; however, the sight of it is enough to disgust or nauseate most people, and it apparently looks painful. This is the result of the yoma flesh implanted inside their bodies.
The most basic of any Claymore's abilities is her enhanced strength, speed and reflexes. A Claymore's sword is incredibly heavy, almost impossible to lift for most humans, but Claymores can easily wield their swords one-handed with no real signs of fatigue. They can move faster than the human eye can see, and can see and hear things that most humans would easily miss. On top of this, they have enhanced regenerative abilities: most fatal wounds will not kill them (the most certain way to kill a Claymore is to take off her head), and any wounds a Claymore sustains can be easily healed as long as she has the energy to do so. They can also reattach severed limbs, and in some cases, regrow limbs that are lost (though Clare, as an offensive type of warrior, does not possess the ability to do this).
All Claymores also have the unique ability to sense yoma. Yoma use a certain kind of energy referred to as yoki, and Claymores can sense this, allowing them to locate yoma that are disguised as humans. Clare is fairly adept at sensing yoki; she is often able to predict the movements of most enemies before they make them, by sensing the flow of yoki to the muscles. This is an ability she picked up from Teresa, but her ability with it isn't anywhere near Teresa's. And as there are very few Claymores in the Gardens, this ability is pretty useless to her now.
In addition to the ability to sense yoki, Claymores can also use it. While yoki is used most of the time to move and fight, it's possible for a warrior to release more of it to give herself a boost to strength, speed and regeneration. At 10% release, a Claymore's eyes change from silver to gold. At 30%, her face becomes twisted and monstrous looking. At 50%, her body changes, becoming much more muscular and more yoma-like in appearance. Past 80%, all control is lost and the Claymore is overwhelmed by her yoma power, transforming into something far worse than a yoma, a monster with an insatiable hunger for human guts known as an Awakened Being. Luckily, most warriors learn through experience when they are beginning to reach this point and will ask a comrade to kill them so that they can die with their humanity intact.
Claymores are also notably non-human in several other ways. They do not need to eat or drink as often as humans--usually, a few bites of food every few days is enough to sustain them, and they can easily go for a week or so without eating. They also need less rest and sleep than humans. Claymores also have the ability to regulate their body temperature (so when in cold areas, they can adjust their body temperature so that they're comfortably warm), as well as the ability to filter what enters their bloodstream. So Claymores are essentially invulnerable to poisons, and can drink as much alcohol as they like while only getting as drunk as they feel like getting.
Clare
As Clare was turned into a Claymore by having Teresa's half-yoma flesh put into her, rather than using the usual method, she's technically only one quarter yoma. Because of this her abilities are weaker than most of her comrades', and she was the lowest-ranked Claymore of her generation (number 47, out of 47 warriors). However, her ridiculous determination to succeed usually means that she can push herself far beyond what is expected of her. It's also been seven years since Clare defected from the Organization, and she spent those years training ruthlessly with the other six ghosts, so she's caught up quite a bit.
Clare also has the following specific abilities:
Quicksword: By releasing all the yoki in her right arm, Clare can cut with great speed and strength -- she's learned to control the rampaging arm to an extent, meaning that she can direct the quicksword at a particular yoki and avoid hitting an ally. The quicksword is so fast that its movements are invisible to the human eye. Clare learned the quicksword from Irene.
Windcutter: Similar to the quicksword, the windcutter allows Clare to use a series of lightning-fast slashes to cut an enemy. This is slower than the quicksword, but more accurate, and it requires no yoki release to use. Clare learned the windcutter from Flora.