"Do you want a slice or a bigger piece?"
Translation:Tu vrei o felie sau o bucată mai mare?
If it's about food, or any portion/fragment that has come off or has been cut, broken etc. from a larger solid, then you say "bucată" in Romanian. So we usually have: bucată=piece.
If it's about a part or spare part of a larger system/machine/machinery, then you say "piesă" in Romanian. Think nuts, bolts, gears and other such doodads. So we usually have: piesă=physical part (of a system). Note that you can also use "piesă" when referring to: a theatrical play (piesă de teatru), a piece of music (piesă muzicală), a piece of clothing (piesă de îmbrăcăminte), a museum piece (piesă de muzeu), a chess piece (piesă de șah), and a few more which I won't cover here.