Fausset's Bible Dictionary
("fiery one"; compare Arabic "burn".) A common title of the Amalekite kings; as Pharaoh of the Egyptian. Numbers 24:7 implies their greatness at that time. Saul's sparing the Agag of his time (1 Samuel 15:32) contrary to God's command, both then and from the first (Exodus 17:14; Deuteronomy 25:17-19), because of Amalek's having intercepted Israel in the desert, so as to defeat the purpose of God Himself concerning His people, entailed on Saul loss of his throne and life. Agag came to Samuel "delicately" (rather contentedly, pleasantly), confident of his life being spared. But Samuel executed retributive justice (as in the case of Adonibezek, Judges 1), hewing him to pieces, and so making his mother childless, as he had made other women childless by hewing their sons to pieces (in consonance with his fiery character, as Agag means). This retribution in kind explains the unusual mode of execution. Haman the Agagite (Esther 3:1-10; Esther 8:3-5) was thought by the Jews his descendant, whence sprung his hatred to their race.
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Fausset, Andrew R. Entry for 'Agag'. Fausset's Bible Dictionary. https://www.studylight.org/dictionaries/eng/fbd/a/agag.html. 1949.