As the name suggests, the Viridian Transformation Stone is a new gizmo that allows you to raise the power level of your weapons, be it a tameable critter or a boss monster. It is a very impressive looking thing to say the least, and is certainly worth the effort if you are into weapon upgrading.
The Viridian Transformation Stone comes in a variety of colors, with the most popular being the emerald green, but it is only available in a limited amount of places (so it is definitely not for everyone). It can be found in Brightstone Ruins, Writheblood Ruins, and the Sellia Hideaway.
It is also possible to get one from the dungeon if you are very lucky, but if you have the patience and the gold to spare then there is no substitute for a good old fashion crafty dungeon run. The Viridian Transformation Stone was not invented by Blizzard, it is a product of many years of design and research. The most effective way to get it is to go through the quest A Letter from Hajoon. The Viridian Transformation Stone is not a requirement for the main event, but it does help you along your quests to achieve the top score.