Classically, alicorn is the term for the horn
of a unicorn
, as the phrase "unicorn's horn" was seen as too cumbersome and the snake oil
salesmen wanted something that sounded exotic
while still being close to what it was supposed to be that they were selling.
Recently, it has also come to mean a winged unicorn. While unicorns with wings are often referred to as pegasi, this is inaccurate, and historically, they were merely reffered to as "unicorns with wings." The practice of calling them alicorns can be found in the book A Swiftly Tilting Planet by Madeleine L'Engle. Piers Anthony has also used this term to describe a winged unicorn in his Incarnations of Immortality series. It is uncertain where and when this usage first came to be.
In 1999, there was a group under the name The Alicorn Project which is attempting to collect references to the word alicorn as a unicorn's horn for an entry into the Oxford English Dictionary. They cite the Odell Shepard work, The Lore of the Unicorn, as the origin of the word, which was based on the Italian word for horn, alicorno. This may be the true origin, but due to the nature of the world of writing, plagiarism often obfuscates history. I am uncertain if this is still an active project.