Noce is the generic word for nut, but it also specifically means walnuts. They are grown all over the country; the only special variety is the noci di Sorrento from Campania.
noce del Brasile: Brazil nuts
noce di burro: a walnut size knob of butter
noce di cocco: coconut
noce di cola: kola nut
noce di macadamia: macadamia nut
noce del Messico: pecan
noce moscato: nutmeg
Image by Giulio