A dewclaw is a vestigial digit of the foot of many mammals, birds, and reptiles (including some extinct orders, like certain theropods). It grows higher on the leg so that in digitigrade species, when the animal is standing, it does not make contact with the ground.
The dewclaws are not dead appendages. They can be used to lightly grip bones and other items that dogs hold with the paws. However, in some dogs these claws may not appear to be connected to the leg at all except by a flap of skin; in such dogs the claws do not have a use for gripping as the claw can easily fold or turn.
There is also some debate as to whether dewclaws should be surgically removed. The argument for removal states that dewclaws are a weak digit, barely attached to the leg, so that they can rip partway off or easily catch on something and break, which can be extremely painful and prone to infection. Hunting dogs for example, can easily tear the dewclaw when running in overgrown vegetation. There is also a slight risk that the dog may scratch and damage the eyes when rubbing its face with its front paws. In other cases, a dog may begin to compulsively lick or bite the dewclaw until it is inflamed, posing risk for infection. Dewclaw removal is usually done when the dog is a puppy, sometimes as young as 3 days old, though it can also be performed on older dogs if necessary (though the surgery may be more difficult then). The surgery is fairly straight-forward and may even be done with only local anesthetics if the digit is not well connected to the leg. Unfortunately many dogs can't resist licking at their sore paws following the surgery, so owners need to remain vigilant.
In some countries removing the dewclaws is illegal, as removing it is said to be unnecessarily painful to the dog, in addition to the claim that the dewclaw will rarely or never suffer injury leading to amputation.
In addition, for those dogs whose dewclaws make contact with the ground when they run, it is possible that removing them could be a disadvantage for a dog's speed in running and changing of direction, particularly in performance dog sports such as dog agility.
There also exists in folklore a story that claims that dogs that have not had their dewclaws removed are immune to snakebite. In America, some pups are commonly sold by breeders "dewclawed", that is with the dewclaws removed (as by a vet) for perceived health and safety reasons. A few breed standards also call for it.
Hoofed animals walk on the tips of special toes, the hoofs. Cloven-hoofed animals walk on a central pair of hoofs, but many also have an outer pair of dewclaws on each foot. These are a little further up the leg than the main hoofs, and similar in structure to them. In some species (such as cattle) the dewclaws are much smaller than the hoofs and never touch the ground. In others (such as deer or pigs), they are only a little smaller than the hoofs, and may reach the ground in soft conditions or when jumping. Some hoofed animals (such as giraffes and modern horses) have no dew claws.