The second banquet is in full swing. As they're sitting around, drinking wine, the king asks Esther if she'd like another gift.
Esther reveals her true identity as a Jew, asking for her life and the lives of her people to be spared.
The king—again, not the brightest bulb in the bunch—asks who the wicked person is who decided to do this thing (even though the king personally signed off on the evil plan, earlier). Esther says that it's Haman.
The king rises in wrath and goes into the palace garden, while Haman—seeing that the king intends to kill him—begs for his life from Esther.
However, he finds no mercy with Esther. When the king returns he sees that Haman has flung himself onto Esther's couch, trying to attack her.
Obviously this doesn't help Haman's case for mercy, so the eunuchs put a bag over his head (or cover his face in some other way) and drag him off to be executed.
They see the giant gallows Haman built to kill Mordecai and decide it would be a pretty handy means of killing Haman… which they proceed to do.