IN this story, Homer Barron is the man that Emily Grierson was supposed to marry (or at least that is what she thought).

Homer was a man who came from up North.  He was in town as the boss of a gang of workers who were working on the roads.  While in town, he started going out with Miss Emily.  She clearly thought that they were going to marry even though he publicly said he wasn't interested in marriage.

She appears to have killed him after he, presumably, told her they weren't getting married.