Figure Analysis
These guys are eunuchs (servants who have been castrated—or are otherwise sexually non-active—so they won't get involved with the women in the king's harem). They don't play a very important role in the story. They're more what some theatrical companies call "Fifth Business": they help the main characters move along in the plot.
Hegai is the chief eunuch in charge of the harem. Esther is able to curry favor with him so that she rises to the top. And Hathach is a eunuch who carries messages between Esther and Mordecai, helping them figure out what to do about Haman's genocide plot.
And that's basically all there is to say about them.