I think you are talking about the "veghar," which is the cowl worn by monks, celibate priests and bishops in the Armenian Church. (The "gh" is pronounced a little like the French "r," but a little smoother.)
Here is a picture of our Catholicos with two Archbishops, wearing their veghars, at a meeting with the governor of California:
Is this the headgear that you are asking about?
In front, it comes down to the eyes, and I am told that it is to protect the monk from temptation. In back it goes down to the waist and, along with the pointed top, that is supposed to protect him from the devil. I once heard my priest say that the top is pointed so demons cannot rest on top of the monk's head. The back is protected because the devil always strikes from behind. At least that is what I have heard. Of course this is all symbolic. Ultimately, it symbolizes the monk's detachment from worldly life. I also heard one person say the pointed top is supposed to represent Mt. Ararat.
It could be there is other symbolism. Like I said, this is just what I have heard.
I'm not sure of the origin; I don't think anyone knows.
Perhaps if vasnTearn sees this, she can give more information.