Agag (אֲגַג) is a proper name, likely a dynastic title (like Pharaoh or Caesar) for the kings of Amalek. The most significant Agag in Scripture is the Amalekite king whom Saul defeated in battle but failed to kill — an act of disobedience that cost Saul his kingdom (1 Samuel 15). The name may mean "high" or "fiery" and appears also in Numbers 24:7 in Balaam's oracle about a coming king of Israel greater than Agag.
The story of Agag is the story of incomplete obedience — and its catastrophic consequences. God commanded Saul through Samuel to destroy Amalek utterly as an act of divine judgment (1 Samuel 15:3). Saul obeyed partially — he defeated the enemy but kept the best livestock and spared King Agag. Samuel's rebuke cuts to the theological core: "To obey is better than sacrifice" (1 Samuel 15:22). Saul's rationalization ("I kept them to sacrifice to God") is the prototype of all religious excuse-making that disguises disobedience. Agag himself was executed by Samuel — "Samuel put Agag to death before the LORD" — but the damage was done. The failure to destroy Amalek would haunt Israel for generations. Haman in the book of Esther is called an "Agagite" — an Amalekite descendant — setting up the near-destruction of all Jews. Incomplete obedience is not partial credit; it is full disobedience with religious decoration.