Modern classical pianists dedicate their careers to performing, recording, teaching, researching as well as learning new works/expanding their repertoire. They generally do not write or transcribe music as pianists did in the 19th century. Some classical pianists might specialize in accompaniment and chamber music while others (relatively few) will perform as full-time piano soloists.
Mozart could be considered the first "concert pianist" as he performed widely on the piano. Composers Beethoven and Clementi from the classical era were also famed for their playing, as were, from the romantic era, Liszt, Brahms, Chopin, Mendelssohn and Rachmaninoff. From that era, leading performers less known as composers were Clara Schumann and Hans von Bülow. However, as we do not have modern audio recordings of most of these pianists, we rely mainly on written commentary to give us an account of their technique and style.
Jazz pianists almost always perform with other musicians. Their playing is more free than that of classical pianists and they create an air of spontaneity in their performances. They generally do not write down their compositions; improvisation is a significant part of their work. Well known Jazz pianists include Art Tatum, Duke Ellington, Thelonious Monk, Oscar Peterson and Bud Powell.
Popular pianists might work as live performers (concert, theatre, etc.), session musicians, arrangers most likely feel at home with synthesizers and other electronic keyboard instruments. Notable popular pianists include Victor Borge who performed as a comedian; Richard Clayderman, who is known for his covers of popular tunes; and singer and entertainer Liberace, who at the height of his fame, was one of the highest paid entertainers in the world.
A single listing of pianists in all genres would be impractical, given the multitude of musicians noted for their performances on the instrument. Below are links to lists of well-known or influential pianists divided by genres: