Alicorn is a term commonly used within the fandom to describe winged unicorns like Princess Celestia and Princess Luna. The show itself does not use the term, referring to the princesses only as "unicorns". Luna is also called a "Pegasus unicorn" on Hasbro's My Little Pony Facebook page.
The term is used in various modern fantasy novels, and extensively so in The Halfblood Chronicles series, however until the late 20th century it was only used to denote the horn of a unicorn or the material it's made of. The word in its new sense is said to be derived of Latin "ala" meaning "wing", and "cornu" meaning "horn", however no sources are cited for this claim.
In fan labor, alicorns/winged unicorns are relatively rare, in keeping with how few of them appear on the actual show. However, Lauren Faust is usually depicted as one, based on a drawing of herself. Original character alicorns include Nyx and (as a parody of authors who insert themselves into their story as an especially powerful pony) Argembarger, the writer of the spiderses. Also, on occasion and likely from her position as Princess Celestia's pupil and a powerful unicorn, Twilight Sparkle may find herself elevated to winged unicorn status.