It's the treasure tool that Yuji keeps inside of his body and that allows him to restore his power of existence every midnight. Evil Denizens are after it because of the immense power that it holds
Reiji Maigo was made by "Engaged Link" Pheles and Yohan. Before Yuji, it was replaced in Yohan. When the "Engaged Link" was confronted by Sabrac, Pheles insert Yohan to the Reiji Maigo to keep him safe and send the Reiji Maigo to random location, but Sabrac inserted a spell that cause Reiji Maigo acting abnormal. It is also revealed that the silver were inside Reiji Maigo. Bal Masque is after Reiji Maigo because they need it to evoke their true leader, Sairei no Hebi.