"Santa Lucia" is a traditional Neapolitan (from Naples) song transcribed by Teodoro Cottrau and published in 1849. It was made in famous in the U.S. from a recording by Enrico Caruso. The video below features tenor Mario Lanza.
Elvis recorded "Santa Lucia" and can be heard in the Youtube video below. His Italian is fairly good.
Free sheet music of "Santa Lucia" (Key of C) is available online at Art Song Central.