Polyphony is the name for that kind of music: several independent tunes woven together at the same time. It is a noun, so you talk about the polyphony in a piece. Western church music from a few hundred years ago is the example teachers reach for first. Though actually, plenty of everyday music has more than one sound at once; people keep this word for when the lines are truly separate, not just a tune plus backing. You will meet it in music class and in serious writing about music, almost never in the street. Do not mix it up with harmony in the loose pop sense — harmony can mean “nice chords,” while polyphony is about several melodies living together.