According to the Grove Dictionary of Music and Musicians, there are two kinds of piano duet: "[pieces of music] for two players at one instrument, and those in which each of the two pianists has an instrument to themselves."
[1] Grove notes that the one-piano duet has the larger repertory, but has come to be regarded as a modest, domestic form of music-making by comparison with "the more glamorous two-piano duet".
Mozart played duets as a child with his sister, and later wrote sonatas for four hands at one piano; Schubert was another composer who composed for the genre, notably with his Fantasy in F minor, D. 940.
Jane Bellingham in The Oxford Companion to Music lists other composers who wrote piano duets, including Brahms, Dvořák, Grieg, Debussy, Stravinsky, and Bartók.
[3] In the late nineteenth and early twentieth centuries French piano duets included Bizet's Jeux d'enfants, Fauré's Dolly Suite and Ravel's Ma mère l'oye.