The Shadow Cabinet is an international society dedicated to keeping the living separated from the dead. They also make sure lay lines stay fairly clear for ghosts to move on from this world to the afterlife. It is secret, and requires the utmost secrecy from its members. Members are living and dead. Not much else is known about it.
[For the third book in the Shades of London series, please see 'The Shadow Cabinet']
Stones
The Cabinet's mission in England is to protect the eight stones responsible for separating the living from the dead. The identity of only two of these stones is known: the Oswulf Stone, and the 12 termini, which were the original Crown Jewels before being shattered.
If a ghost touches one of these stones they are destroyed in a flash of light. No one knows if they go to an afterlife or not.
The Cabinet & Shades
According to Stephen, there is always one member of the Shades who is also a part of the Cabinet. In this case, that is him. Currently he is the only one who knows that it is an existing thing, while the other characters all believe it is a conspiracy theory. He pretends to believe so as well. The Shadow Cabinet either recruited him because of his discretion, or his place as Shades leader, or maybe even both reasons. He said that they approached him early on in his career, but it's unknown if that was before he met Boo (his first recruit) or after.
In Practice
Despite their benevolent motive, the Shadow Cabinet itself does shady deeds to succeed in their mission, such as following and watching potential recruits, presumably without their knowledge, before approaching them and explaining bits and pieces to him/her. Although it's unknown what would happen if a Shade declined to join them, once he or she does, it becomes their top priority, and they can't quit.
History
The oldest known mention of the Cabinet is in reference to Thomas Blood's attempt to steal the Crown Jewels in 1671, but it is believed to be older than that.
There are many rumors about the Shadow Cabinet online. Stephen says that the most ridiculous ones are actually created by the Cabinet itself to make it seem unreal. As a member himself, Stephen knows about other especially spiritual cities: Paris, Giza, Rome, Shanghai, Baghdad, New York, Santiago, New Delhi, Sao Paulo, Alexandria, New Orleans, and unnamed others. These places are believed to be more spiritually active because they are near large bodies of water, a spiritual magnet. It can also be assumed that they each have their own branches of the Cabinet.
According to Thorpe, the British government has no knowledge of the group, but it can be speculated that his higher-ups might. They intentionally share a name with a government Cabinet to cause confusion.
For some reason, the roman numerals one through twelve, with two repeating sevens, is significant to them.