Aai means came or has come. However "chord aayi" means "left behind""Pichhli gali mein jaane kya chhod aayi main" literary means "I don't know what I left behind in the alley or lane behind".Metaphorically she is probably talking about memories (probably painful past) that she left behind.though I do not remember the complete song nor have I seen the movies

It means to" come in" (to a female) While aaye is the respective way of saying "come"