The Alicorn has no specific name, and has been called different names like a Winged Unicorn, Cerapter (keros=horn + pteros=wings), Unisis, Pegacorns, Hornisis, Unipegs and Hornipegs. The horn of a unicorn that has miraculous healing powers is called an 'Alicorn'.
The term 'Alicorn', which became popular in My Little Pony (see Popular Culture below), was made up of the French words Aile (wing) and Corne (horn).
Alicorns are seen and depicted in art. The ancient Achaemenid Assyrian made Alicorns a representation of evil and darkness, whereas sometimes they are seen to be representations of light, or sided with light. An Irish poet W.B Yeats wrote of a winged best he associated with laughing, ecstatic destruction. Later the beast took the form of a Winged Unicorn in his play 'The Unicorn from the Stars'. The poem 'The Second Coming' by Bethlehem also slouched towards having the winged beast.
To most Asian cultures there is no difference between a Winged Unicorn and a regular Unicorn (and Pegasus don't exist in that culture).
Depending on the specific culture unicorns either don't exsist in their folklore or all unicorns have wings. Either that or the unicorns can make their wings appear and dissapear at their own will, and are usually portrayed as Sapient Steeds (talking horses).
Appearances in Modern Culture
My Little Pony
In My Little Pony by Hasbro Studios there are portrayed unicorn cross pegasus called Alicorns. The Alicorns have the magical unicorn abilities and grace and flying abilities of pegasus, and are usually seen as royalty. The Alicorns have the most important jobs, like raising the sun, the moon or ruling over towns and cities.
In Fire Emblem series, a Nintendo published video game, members of the "Falcon Knight" class are seen to be riding winged unicorns.