It's called counterpoint. Two or more melodic lines of more or less equal importance sounding simultaneously. As opposed to homophonic textures, which feature one melody with harmony and/or accompaniment (or, as my friend says, "I'm not homophonic- I like gay people.").
The quartet Johanna from Sweeney Act II gets me every time.
A Weekend in the Country from Little Night Music
The Act I finale of West Side Story
Aiutami from Piazza
"WHEN is the winter of our discontent?"
"NOW is the winter of our discontent!"
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