Press is a surname with two unrelated origins.
In England and Wales, it derives from Priest or Price.
In Eastern Europe (especially centered around Minsk), it is a Jewish name, likely derived from the Sephardic surname Peres (Perez, Peretz, Perutz).
[1] It also is a metonymic occupational surname that refers to the occupation of someone who ironed clothes, derived from the Yiddish "pres" or flat iron.
[2] People with the surname include: