I have also heard it called the mandylion, but I don’t know what that term means
"Mandylion" comes from the word “mantle” (MANDYAS in Greek).
According to the story, King Abgar V Ouchama (AD 13 - 50) of Edessa wrote a letter to Jesus, asking him to come cure him of an (incurable) illness (which some identify as leprosy), which he sent via his secretary, Hannan (to which he was also said to have entrusted a task of painting an image of Jesus):
"*Abgar Ouchama to Jesus, the Good Physician who has appeared in the country of Jerusalem, greeting:
I have heard of you, and of your healing; you do not heal by medicines and by drugs, but by your word you make blind men see, make the lame walk and the lepers clear, and you make deaf men hear and you cast out spirits and night-wandering demons, and you heal the miserable by your word, and also raise dead men to life.
I heard these great wonders that you do, I put in my mind that either you are God who came down from Heaven and did these, or you are the Son of God, that you do all these.
I write and ask you that you come to me, that I may prostrate myself before you, and you heal a certain pain that I have, since I believe in you. Also, I have further heard this, that the Jews are murmuring against you and persecuting you, and even wish to crucify you and intend to do away with you. I have but one small city, but it is beautiful and sufficient for the two of us to inhabit in peace.*"
When Hannan came to Jerusalem, he attempted to paint Jesus’ image, to no avail. Jesus then saw and called him by name (Hannan was standing atop a rock as there are a lot of people around Jesus), and asking for a towel and a basin of water, He washed His face and rubbed it on the towel and an image of His face appeared. He then dictated his reply to Hannan and sent him back to Abgar along with the towel and the letter:
"*Blessed are you that when you have not seen me you believe in me! For it is written about me that those who see me will not believe in me, (and those who do not see me will believe and live).
And that you have written to me that I should come to you that for which I have been sent here is now fulfilled, and I am about to ascend to my Father who sent me; and when I have ascended to him, I will send you one of my disciples, who will heal and cure whatever pain you have, and all who are with you he will lead to eternal life (and your city will be blessed, and no enemy in the future will ever take it over).*’"