The metal is an alloy called adamantium, supposedly indestructible. It's grafted onto his entire skeleton.

Jean Grey

Weapon X was the title given to the tenth subject to the experiments done by William Stryker. The purpose of the experiment was to figure out if a Mutant could successfully be bonded with Adamantium. The subject of the experiment was the Wolverine, however, it was only successful due to his healing factor. Hence, all bonding experiments done with Adamantium thereafter were not full-skeleton bonds.