The image most people carry, the white beast laying its head in a virgin's lap, comes from one text and one purpose.
The Physiologus, a Greek Christian compilation of late antiquity that became one of the most widely translated books of the Middle Ages, describes each animal and then reads it allegorically. Its unicorn is small, fierce and impossible for hunters to take; it is caught by seating a virgin in its path, whereupon it comes to her, lays its head in her lap and is captured. The allegory is stated outright: the unicorn is Christ, the horn is his power, the virgin is Mary, and the capture is the Incarnation. This is not a folk tale that acquired a Christian gloss. It was composed as a Christian gloss.
From the Physiologus the scene passes into the bestiaries and out of them into tapestry, of which the Cluny and Cloisters hunt cycles are the famous survivals, and into heraldry, where the unicorn is chained, a detail that only makes sense if you know it cannot otherwise be held.
The alicorn trade is the strangest chapter. Powdered or whole unicorn horn was sold in Europe as an antidote to poison, and cups reputedly made from it were royal treasures; sums involved were enormous, and testing food with alicorn was a court practice. Most of what was sold was narwhal tusk, brought south by Scandinavian traders who had good reason not to explain their supply. The Danish physician Ole Worm demonstrated the identification in 1638, and yet false alicorn powder was still being sold medicinally in Europe as late as 1741.
The unicorn remains in the royal arms of the United Kingdom, where the Scottish unicorn has stood beside the English lion since the union of 1707.