It is a Spanish word that means "little daddy", but is a term of endearment, a loving nick-name. That is how "Papi" is used in the song. This is just a word between the singer and her lover...no one else would call him that.
An old fashioned way to say "Daddy" is Papa, and this is the correct word in Spanish also.
One hundred years ago many American children called their father "Papa", and that is where the word comes from.