The cat claw extends from the P3 bone, and it is analogous to the human fingernail extending from the tip of our finger. The excisional method removes the entire bone (P3), and is most commonly done using a scalpel blade. Some vets use laser and report less hemorrhaging. The guillotine method uses a nail clipper than severs the P3 bone in half, removing the claw and distal (end) part of this bone, sometimes cutting through the toe pad as well.
A third technique, called a Deep Digital Flexor Tendonectomy, is not technically a declaw, but a surgical procedure where the tendons that operate the claw are cut; leaving the claw intact. This is to prevent active use of the claws. Care must be taken after this procedure to keep the nails trimmed, so they do not grow into the pad (the cat doesn't have control over the claw).
A few comments: the excisional method takes more time, and thus may cost more. I feel that cutting through the bone (guillotine method) creates a more painful recovery with more possibility for complications. Pet owners should be aware of alternatives to declawing, such as: behavior modification, application of Soft Paws (or similar) and regular nail trimming to make an informed decision.
Related Reading:
Declawing & Humane Alternatives
From Cats Guide, Franny Syufy who has many declawing resources on her site to check out.
The Declawing Dilemma... what you should know about declawing
In-depth look at declawing.
Declawing in Cats
From the American College of Veterinary Surgeons (ACVS), with surgical photos.
The Declawing Debate
From Sue A. Whitman, DVM.
