The correct term for an arthropod's 'claw' is a chela (plural chelae). Legs bearing a chela are called chelipeds. Chelae are also called pincers.
A talon is the claw of a bird of prey, its primary hunting tool.
In tetrapods, claws are made of keratin and consist of two layers. The unguis is the harder external layer, which consists of keratin fibers arranged perpendicular to the direction of growth and in layers at an oblique angle. The subunguis is the softer, flaky underside layer whose grain is parallel to the direction of growth. The claw grows outward from the nail matrix at the base of the unguis and the subunguis grows thicker while travelling across the nail bed. The unguis grows outward faster than the subunguis to produce a curve and the thinner sides of the claw wear away faster than their thicker middle, producing a more or less sharp point. Tetrapods use their claws in many ways, commonly to grasp or kill prey, to dig and to climb and hang.
A nail is homologous to a claw but is flatter and has a curved edge, instead of a point. A nail that is big enough to bear weight is called a 'hoof' (see also Horse hoof. However, one side of the cloven-hoof of artiodactyl ungulates may also be called a claw).
Every so often, the growth of claws stops and restarts, as does hair. In hair, this results in the hair falling out and being replaced by a new one. In claws, this results in an abscission layer and the old segment breaks off. This process takes several months for human thumbnails. Cats are often seen working old unguis layers off on wood or on boards made for the purpose. Ungulates' hooves wear or self-trim by ground contact. Domesticated equids (horses, donkeys and mules) usually need regular trimming by a farrier, as a consequence of reduced activity on hard ground.
The Human nail consist of the unguis alone; the subunguis has disappeared. Much like the tail, the claw in humans is no longer necessary. However, in some individuals the subunguis has redeveloped to form a claw-like structure. These individuals don't have actual claws, instead the unguis appears normal with the addition of a thick growth of subunguis. The texture might be described as "woody". 
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