I am reading about the Israelites time in the desert during their exile from Egypt. What did God mean when He said (Exodus 17:14): "I will blot out the memory of Amalek from under the heaven"?
Numbers 24:20, which gives Balaam's fifth oracle, says:
Then Balaam saw Amalek and spoke his message:
“Amalek was first among the nations, but their end will be utter destruction.”
As pointed out above, the reference to blotting out the memory of Amalek is further repeated in Deuteronomy 25:17-19 and makes clear there that God will do this through the children of Israel after they have entered the land of Canaan:
17 Remember what the Amalekites did to you along the way when you came out of Egypt. 18 When you were weary and worn out, they met you on your journey and attacked all who were lagging behind; they had no fear of God. 19 When the Lord your God gives you rest from all the enemies around you in the land he is giving you to possess as an inheritance, you shall blot out the name of Amalek from under heaven. Do not forget!
I Samuel 15 shows how this command was (partially) fulfilled in the land of Israel through the agency of Saul:
15 Samuel said to Saul, “I am the one the Lord sent to anoint you king over his people Israel; so listen now to the message from the Lord. 2 This is what the Lord Almighty says: ‘I will punish the Amalekites for what they did to Israel when they waylaid them as they came up from Egypt. 3 Now go, attack the Amalekites and totally destroy all that belongs to them. Do not spare them; put to death men and women, children and infants, cattle and sheep, camels and donkeys.’”
4 So Saul summoned the men and mustered them at Telaim—two hundred thousand foot soldiers and ten thousand from Judah. 5 Saul went to the city of Amalek and set an ambush in the ravine. 6 Then he said to the Kenites, “Go away, leave the Amalekites so that I do not destroy you along with them; for you showed kindness to all the Israelites when they came up out of Egypt.” So the Kenites moved away from the Amalekites.
7 Then Saul attacked the Amalekites all the way from Havilah to Shur, near the eastern border of Egypt. 8 He took Agag king of the Amalekites alive, and all his people he totally destroyed with the sword. 9 But Saul and the army spared Agag and the best of the sheep and cattle, the fat calves and lambs—everything that was good. These they were unwilling to destroy completely, but everything that was despised and weak they totally destroyed.
10 Then the word of the Lord came to Samuel: 11 “I regret that I have made Saul king, because he has turned away from me and has not carried out my instructions.” Samuel was angry, and he cried out to the Lord all that night.
12 Early in the morning Samuel got up and went to meet Saul, but he was told, “Saul has gone to Carmel. There he has set up a monument in his own honor and has turned and gone on down to Gilgal.”
13 When Samuel reached him, Saul said, “The Lord bless you! I have carried out the Lord’s instructions.”
14 But Samuel said, “What then is this bleating of sheep in my ears? What is this lowing of cattle that I hear?”
15 Saul answered, “The soldiers brought them from the Amalekites; they spared the best of the sheep and cattle to sacrifice to the Lord your God, but we totally destroyed the rest.”
16 “Enough!” Samuel said to Saul. “Let me tell you what the Lord said to me last night.”
“Tell me,” Saul replied.
17 Samuel said, “Although you were once small in your own eyes, did you not become the head of the tribes of Israel? The Lord anointed you king over Israel. 18 And he sent you on a mission, saying, ‘Go and completely destroy those wicked people, the Amalekites; wage war against them until you have wiped them out.’ 19 Why did you not obey the Lord? Why did you pounce on the plunder and do evil in the eyes of the Lord?”
20 “But I did obey the Lord,” Saul said. “I went on the mission the Lord assigned me. I completely destroyed the Amalekites and brought back Agag their king. 21 The soldiers took sheep and cattle from the plunder, the best of what was devoted to God, in order to sacrifice them to the Lord your God at Gilgal.”
22 But Samuel replied:
“Does the Lord delight in burnt offerings and sacrifices as much as in obeying the Lord? To obey is better than sacrifice, and to heed is better than the fat of rams. 23 For rebellion is like the sin of divination, and arrogance like the evil of idolatry. Because you have rejected the word of the Lord, he has rejected you as king.” 24 Then Saul said to Samuel, “I have sinned. I violated the Lord’s command and your instructions. I was afraid of the men and so I gave in to them. 25 Now I beg you, forgive my sin and come back with me, so that I may worship the Lord.”
26 But Samuel said to him, “I will not go back with you. You have rejected the word of the Lord, and the Lord has rejected you as king over Israel!”
27 As Samuel turned to leave, Saul caught hold of the hem of his robe, and it tore. 28 Samuel said to him, “The Lord has torn the kingdom of Israel from you today and has given it to one of your neighbors—to one better than you. 29 He who is the Glory of Israel does not lie or change his mind; for he is not a human being, that he should change his mind.”
30 Saul replied, “I have sinned. But please honor me before the elders of my people and before Israel; come back with me, so that I may worship the Lord your God.” 31 So Samuel went back with Saul, and Saul worshiped the Lord.
32 Then Samuel said, “Bring me Agag king of the Amalekites.”
Agag came to him in chains. And he thought, “Surely the bitterness of death is past.”
33 But Samuel said,
“As your sword has made women childless, so will your mother be childless among women.” And Samuel put Agag to death before the Lord at Gilgal.
34 Then Samuel left for Ramah, but Saul went up to his home in Gibeah of Saul. 35 Until the day Samuel died, he did not go to see Saul again, though Samuel mourned for him. And the Lord regretted that he had made Saul king over Israel.
I Samuel 27:8 makes clear that Saul and his men did not kill all the Amalekites, for we find there David killing some in Ziklag:
27 But David thought to himself, “One of these days I will be destroyed by the hand of Saul. The best thing I can do is to escape to the land of the Philistines. Then Saul will give up searching for me anywhere in Israel, and I will slip out of his hand.”
2 So David and the six hundred men with him left and went over to Achish son of Maok king of Gath. 3 David and his men settled in Gath with Achish. Each man had his family with him, and David had his two wives: Ahinoam of Jezreel and Abigail of Carmel, the widow of Nabal. 4 When Saul was told that David had fled to Gath, he no longer searched for him.
5 Then David said to Achish, “If I have found favor in your eyes, let a place be assigned to me in one of the country towns, that I may live there. Why should your servant live in the royal city with you?”
6 So on that day Achish gave him Ziklag, and it has belonged to the kings of Judah ever since. 7 David lived in Philistine territory a year and four months.
8 Now David and his men went up and raided the Geshurites, the Girzites and the Amalekites. (From ancient times these peoples had lived in the land extending to Shur and Egypt.) 9 Whenever David attacked an area, he did not leave a man or woman alive, but took sheep and cattle, donkeys and camels, and clothes.
That there were still more Amalekites elsewhere we find from I Samuel 30:1-2, where we are told that the Amalekites have taken revenge on David for his slaughter of their people in Ziklag:
30 David and his men reached Ziklag on the third day. Now the Amalekites had raided the Negev and Ziklag. They had attacked Ziklag and burned it, 2 and had taken captive the women and everyone else in it, both young and old. They killed none of them, but carried them off as they went on their way.
I Chron. 4:42-43 states that in the time of Hezekiah five hundred Simeonites annihilated the last remnant "of the Amalekites that had escaped" on Mount Seir and settled there in the place of Amalek.
On Purim, observant Jews drink to celebrate the blotting out of the nation of Amalek, of whom Haman is said to be a descendant (Esther 3:1).
Reform Judaism considers the ban against Amalek a primitive element in Tanakh and unacceptable for modern society. (See Gunther Plaut, editor. The Torah: A Modern Commentary (2005); Rabbi Simeon J. Maslin, 'Purim, the Bible, and A Vengeful God,' Reform Judaism.org (2/25/2015): https://reformjudaism.org/.../purim-bible-and-vengeful-god). For the views of orthodox Judaism, Rambam among them, see Shmuly Yankowitz's article in the bibliography below.
When all of the above passages in the Tanakh are collated, we discover that 'I will blot out the memory of Amalek' means 'I will utterly destroy Amalek' (Numbers 24:20) 'through you, the children of Israel' (Deuteronomy 25:19), 'I will totally destroy them (I Samuel 15:3), wipe them out (1 Samuel 15:18). I place them under cherem (decree of destruction), as I have the seven nations of Canaan.' In modern terminology, 'I sanction their genocide by you the children of Israel.'
Bibliography: Lerner, Berel Dov. 'Saul and Genocide.' Jewish Bible Quarterly 42:1 (January-March, 2014) 39-44: https://www.questia.com/library/journal/.../saul-and-genocid...; Shmuly Yankowitz, 'Genocide in the Torah,' My Jewish Learning: https://www.myjewishlearning.com/.../genocide-in-the-torah...; Shaul Magid, 'The Dark Side of Purim,' The Forward (March 10, 2014): https://forward.com/opinion/194161/the-dark-side-of-purim/.