The answer to the "who" part of this question is the infamous monk, Grigori Rasputin.
I suppose that the most likely answer to the "what" part of the question would be religion, or perhaps in Rasputin's prayers and his claims that he was able to heal Alexei through those prayers.
The whole royal family, but especially Alexandra, put a great deal of trust in Rasputin, who apparently did actually manage to help Alexei to some degree. There is a great deal of debate over how he did this.
Because of his apparent effect on Alexei, Rasputin got to be quite powerful before he was assassinated.