“Haman.” Haman was “the enemy of the Jews” (Esther 3:10; 8:1; 9:10, 24), and a representative and prophetic picture of the Devil himself (see commentary on Esther 5:14).
“the Agagite.” It has long been believed by both Jews and conservative Christians that Haman is a descendant of Agag, who was a king of the Amalekites at the time of Saul (1 Sam. 15:8-9, 32-33). Although there is no way to prove that, God is the Author of Scripture (2 Tim. 3:16), and one of the purposes of the Bible is to inform us of things we need to know to understand how evil and evil spirits can work through many generations. Given that, what Jews and Christians have believed through the centuries about Haman being a descendant of Agag makes sense and fits with the overall purpose of Scripture.
We know that when Saul killed the Amalekites as recorded in 1 Samuel 15, he did not kill all of them because there were still Amalekites during the reign of David (1 Sam. 27:8; 30:1). Since Agag was the king of the Amalekites during the time of Saul, it makes sense that like most kings he would have had more than one wife and would have had quite a few sons. Those sons were princes, and most likely at least some of them would have acted as under-rulers to Agag and lived in cities scattered around the Amalekite kingdom. That explains why when Saul killed the Amalekites as recorded in 1 Samuel 15, that some of the sons of Agag survived—they were not in the cities that Saul attacked. Furthermore, those sons had descendants who then survived through the reigns of David and other Judean kings, and eventually moved into other areas of the ancient Near East such that by the time of Esther, which was some 500 years after Saul, there were descendants of Agag, Agagites, in the Persian empire. Add to that the fact that it is common in the Eastern culture for people to have very long historic memories, including blood feuds that last for generations, and we can see why when Haman got angry at Mordecai he did not just want to kill Mordecai, but he wanted to use the occasion to exterminate all the Jews. Actually, Mordecai felt the same way about Haman as Haman did about Mordecai. Mordecai would not bow down before Haman because Mordecai was a Jew and he knew the Amalekites were enemies of God (Esther 3:4).