The Birdcage is a small department in the British Intelligence community.
It is not officially called the Birdcage. Officially it really does not have a proper name. It just is. When we first hear of it, it is rather in the shadows, sometimes referred to as the Department. It initially comes across a somewhat stern and cold but it obviously needs to be because it is working on behalf of the British Government and no particular person so it can not really have a heart. It only as time goes by and the people working for the Department, which they might occasionally refer to as the Birdcage, become a little bit less shrouded in the shadows that we begin to see there is something decidedly mean about it.
The Birdcage is not really a bad organization in that it does not go out of its way to break the laws of the nation, except of course when it is expedient to blackmail or corrupt or main or even kill to get the job done. Even then it likely would talk about the "greater good" and "best interests" and so on. But in the case of some of the people whose lives are messed with if not actually snuffed out, it would be beyond most people to agree with the actions.
The stories told in the series run a huge gamut of plots and characters. Most do not have recurring people in them, even if the people from one book might survive the goings-on. There are a few people from the Department that might show up in an adventure but not the next. People like Quint who is the one to call if accidents need happening. Or Kerslake who always seems to have a lot more knowledge of what is going on than he should and that is scary. Or Warboys who does not seem to be the top dog but perhaps really is.
These occasional people are good for putting some faces to the people behind the scenes but other cases there will be no sign of them - just the increasingly more sinister aspect of an agency with very little real oversight and the ability to make its own decisions about what is right and wrong.