Skeletal animation is a modelling technique in which, in general, a polygonal or other solid mesh is connected and deformed by a collection of joints and bones.

A bone may be connected to other bones, via joints. A bone may have one 'parent' joint, and several children. Each bone's angle and position will be computed in relation to its parent, and the parent of that, etc, until there is no parent.

Bones consist of a length, an angle (or quaternion), and children bones. The drawer will start at the root bone, rotate the vertices/part of mesh affected by the bone. It will then draw the bone's children, in relation to the parent bone, in the same manner. The result of this is a new mesh, that is deformed to fit the bone structure.

The advantages of this, over simply altering the position of vertices in pre-set fashions (such as in Quake) are smoother animation playback, easier animation creation, and the ability for dynamic movement of limbs.
Dynamic animation, such as ragdoll physics, where a man caught in an explosion could fly away from the impact, and react to the environment as if his muscles were completely relaxed, is possible. As is blending of different animations, to create a seamless rendition from walking to running, or talking and walking.

Here is some pseudo code, to describe the basic idea of writing a skeletal animation renderer:
function bone($bone, $matrix)
  rotate $matrix by the angles of $bone
  add vertices in $bone to new array, rotating each by $matrix
  translate $matrix by the length of $bone
  for each child of $bone, recurse this function with the matrix we now have and add vertices to the array
  return the array of vertices

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