A transposing piano is a special piano with a mechanism (operated by a pedal or lever) that changes the keyboard position relative to the action (see Development of the modern piano for details). This transposes (changes the key of) any particular keyboard fingering.
Transposing pianos let people who only play well in certain keys use familiar fingerings to play in other keys—and let players who are proficient but can't transpose on the fly spontaneously accommodate different keys. Transposing pianos were never common, and few still exist.
Irving Berlin had two such instruments. One was donated to the Smithsonian Institution in 1973, and is on display in the National Museum of American Jewish History. Berlin never learned to read music, playing his songs entirely by ear in the key of f-sharp, employing his “trick piano” to do the work as necessary.
Individuals with absolute pitch may have difficulty playing on such a piano, because the pitches they actually hear do not match the notes they are playing on the keyboard. Some such people can overcome the difficulty with practice, however.
Many electronic or digital pianos and keyboards can transpose. The harmonium sometimes features a mechanically-shifted keyboard for transposition.