You can either continue saying ‘the white part of the egg,’ and the ‘yellow part of the egg’ in Italian, or you can take this opportunity to be fancy and use, albume and tuorlo.
Albume = egg white (m. noun)
Tuorlo = egg yolk (m. noun)
Albume comes from the Latin, albus, or white. (I guess that’s where albino comes from too.)
Tuorlo comes from the Latin, torulus, meaning a round perturbation, much like the round yolk of an egg.
Italy may not be at the forefront of egg dishes or omelette-ry, so you’re probably not going to be asking for an egg white anything, but it’s still good to know… for like cake purposes maybe…
‘Mangio solo l’albume dell’uovo, perchè il tuorlo non è sano.’ – I only eat the egg white, because the yolk is not healthy.