The term is Greek for “Not handmade", but these items are more commonly known as “Icons not made by human hands.” This is in reference to a particular type of icon which just appears through some sort of divine miracle, either on an object or in a heavenly vision.
Usually the images are of either Christ or the Virgin Mary, and are often highly revered by the faithful, as they functioned not only as icons but as relics as well. They were seen as important references for other images within icon art, as they were (and still are) thought to be reliable sources of how the subjects actually looked. They were copied to an extensive degree, because Byzantine tradition dictates the conservation of the true form of the subjects. Surviving icons today are notably similar in appearance due to this idea.
Some noteworthy acheiropoieta:
Veil of Veronica:
A legendary Christian relic thought to have miraculous properties such as restoring sight to the blind, quenching thirst and raising the dead. Legend has it that Veronica from Jerusalem encountered Jesus on her travels, and when she stopped to wipe his sweat from his brow with her veil, an imprint of his face was left on the veil. Veronica later traveled to Rome and presented it to Tiberius, the Emperor of Rome.
More commonly known to the West as the Image of Edessa, the Mandylion was a holy relic on which the face of Christ was miraculously imprinted onto a rectangular piece of cloth. The earliest documentation of the image is from the 6th century, from the city of Edessa (known now as Urfa). In the 10th century, it was moved to Constantinople, but Crusaders stole it when they sacked the city in 1204. It reappeared in La Sainte-Chapelle in Paris, but disappeared for good during the French Revolution.