How do demotions work in the Catholic Church? All other controversial materials aside, as I read this article about Raymond Burke, I'm curious about this line, and this line only (emphasis mine):


  Burke, who was recently demoted from the Vatican’s highest court to a
  ceremonial philanthropic post, also pointed to the introduction of
  altar girls for why fewer men are joining the priesthood.


I'm just curious, how do such demotions work?  Are they unilateral decisions from the Pope?  Is there some other established methodology for determining status in the Church?
In this particular case the word demotion is misleading. Better word would be transfer.  It might be seen as a demotion for an outsider, but in reality it is not. He is still a Cardinal and a Bishop. You have to separate the Catholic Church's spiritual hierarchy from its functional hierarchy. Within functional hierarchy, heading the roman curia is to help the pope in his work. A cardinal has to do what the pope wants him to do. Every Cardinal vow obedience to the pope. They know that such a thing can happen.

Are they unilateral decisions from the Pope?

Theoretically it is an unilateral decisions of the Pope. But practically it is done by the pope after consulting other Cardinals and bishops.

Is there some other established methodology for determining status in the Church?

By status if you mean becoming a Priest, Bishop, Cardinal or a Pope, then yes, there is. This video gives you a rough outline of that procedure. Remember Holy Orders leaves an indelible mark on the soul, Hence once a priest always a priest (Even in Hell).