In the first book, the artifact of choice was the Mortal Cup. This time, Valentine is after the second of the Mortal Instruments: Maellartach, the Soul-Sword, the Angel Blade, the Mortal Sword. Can't they just pick one name and stick to it? Jeez.
Oh, and this is "the blade with which the Angel drove Adam and Eve out of the garden" (1.5.60) so it's been around a while.
The Clave keeps the Soul-Sword (that's the easiest to spell) around for the Inquisitor's use. She wields the sword to determine if a Shadowhunter is lying, and maybe to chop vegetables if all the other cutlery is dirty.
Valentine claims the Soul-Sword for himself fairly easily and early on in the book. Why does he want it? Why to do the Ritual of Infernal Conversion (1.8.88) of course, a procedure that involves heating the sword until it's red-hot then cooling it in blood. Not just any blood, but the blood of a Downworlder child—vampire, werewolf, warlock, or faerie. Why do demonic rituals always have to be so violent?
Having an evil Soul-Sword would allow Valentine to summon demons any time he wants, which is a lot easier than putting an ad on demonfriendfinder.com. And that's the interesting thing, symbolically, about this crazy Soul-Sword. There's a pretty clear link between Adam and Eve's shenanigans in the Garden of Eden (they lied about eating the dang fruit), getting kicked out via Soul-Sword threatenings, and the Inquisitor using the sword to tell if people are lying. It seems, symbolically and practically, like a truth-telling device. Good, right?
Except it also summons demons. The Soul-Sword is a reminder that all things can be used for both good and evil in both Clare's twisted world (and our only slightly less twisted real world).
At the end of the book, Valentine still has the sword and the cup in his possession. Which artifact will he claim next? The Mortal Spoon? The Mortal Whisk? We'll have to wait until the next book to find out.