What and who did Alexandra put her faith in to deal with Alexei's illness?
Rasputin is a fairly interesting character. Medical science at the time was not enough nor capable enough for Alexandra, as she believed Rasputin possessed powers to stop Alexei's hemophiliac bleeding spells. She believed that Rasputin's powers were sent from God and worthy of all of her support and faith. Rasputin was used to be another example of how the Czar was out of touch with reality, as he allowed a perceived "fraud" to enter the Royal Court. Some also argued that Rasputin exercised a legislative and political role in the Royal Court. The exact role that he held in this regard is not clear, but his presences was used and manipulated by those who stood in opposition to the Czar.
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.