Can anyone tell me what the "keeper" inside Yuji Sakai is? I'm watching the anime (second season) and I'm about to finish it soon, but I still don't get the meaning of the phrase "keeper".
It's the Reiji Maigo.
It's a tool that was created between the two lovers (Johann and Pheles) to grant eternal life to those that are not Flame Haze. This is because:
Johann is a Mystes, while his lover is not. They wanted their life to last forever together, so this was created to allow Johann to live forever. However, this backfires a bit later as you will discover in the series. In the end, after the defeat of Sabrac and the ending of Shakugan no Shana, this resolves itself a bit, though.
The Silver and Reiji Maigo is a powerful entity that the Ball Masque want. They attempt to use it later in Season 3 to create a gateway to another world for all the "Guze no Tomogara" (Crimson Denizens). There is nothing particularly special about Yuji having this except that he was randomly endowed with it when it was sealed away from Johann.
This is due to Sabrac, who is later found part of Ball Masque and killed.
All in all, the keeper is a source of infinite existence that the Ball Masque want to create Xanadu.