"Remember what Amalek did to you on your way out of Egypt - how he chanced upon you on the way, striking all the weakest among you from behind, when you were tired and worn out, and he did not fear God. And when HaShem your God will grant you respite from all your surrounding enemies, in the Land which HaShem your God gives you as an inheritance to posses it, you will eliminate the memory of Amalek from under the heavens, you shall not forget." (Deuteronomy 25:17-19)
These three versus contain three practical mitzvot, two positive and one negative: to remember what Amalek did to us; to exterminate his seed; and not to forget what he did to us (Sefer HaChinuch, mistwot 603, 604, 605; Rambam, "Laws of Kings" 1:1, 5:5, 6:4; Sefer HaMitswot, positive mitswot 188, 189, negative mitswot 59).
The essential question arises: Why does the Torah single out Ama
The seven Canaanite nations were no less implacable enemies of ours.
lek for special treatment? After all, the seven Canaanite nations were no less implacable enemies of ours, yet we are not commanded to exterminate them. Those nations have the option of surrendering and living with us in peace, in Israel, or of leaving Israel unharmed. In either event, we no longer have any animosity towards them (see the Rambam, "Laws of Kings" 6:5, based upon the Jerusalem Talmud, Shvi'it 6:5; see also Tosafot to Gittin 46a, s.v. "keivan de-amru").

The seven Canaanite nations were no less implacable enemies of ours.

The Torah specifically enjoins us to abhor neither an Edomite nor an Egyptian (Deuteronomy 23:8), in spite of the Edomite's hostility to us and the Egyptian's genocidal persecution; and even the open physical and spiritual warfare of the Ammonites and Moabites earned no more opprobrium than the prohibition to intermarry with them even if they converted to Judaism (ibid. v. 4; Rambam, "Laws of Forbidden Relationships" 12:17-18; Shulchan Aruch, Even HaEzer 4:2).
It is only Amalek who can never convert, who has to be exterminated, and whom we have to hunt down even outside of Israel.
The Ohr HaChaim points out what was unique about Amalek: "There was no one like him, who attacked Israel without warning, like a soaring eagle." (Deuteronomy 25:17) And, on the phrase "when you were tired and worn out," he develops the theme: "The Torah points out the causes that make him so hated by us: he caused Israel terrible distress by attacking them like a soaring eagle, without warning, which is what the expression 'chanced upon you' connotes. They were tired and worn out, and had no inner strength left to draw on with which to fight, which was not the case in any of the other wars."
What singled out Amalek was that his attack on us in the desert (Exodus 17:8-17) was totally unprovoked, motivated purely by hatred for hatred's sake.
The hideously cruel Egyptian persecution had at least some rationale: in the early stages, the Egyptians wanted cheap slave labour; the murder of the baby boys was a response to their fear that the Jews might overwhelm Egypt by sheer numbers; and underlying everything was their fear that the Jews could turn against their Egyptian masters in war (Exodus 1:9-10). Not a logical fear, certainly not a justified fear - but a very real fear nonetheless.
Similarly, the Ammonites and the Moabites fought against us in the desert in order to protect their own territorial integrity. For sure, they were punished for their blinding selfishness in refusing us passage through their territory, but nonetheless, one can understand a nation closing its borders to another nation that wants to enter their territory. And the seven Canaanite nations, too, fought against us to protect what they thought of as their land. True, they were relatively new in Israel when we returned; true, they had neither historical nor legal claim to Israel; nonetheless, they were fighting for their homes. A fight like that, even though those nations were in the wrong, can be understood.
Even the countries who would invade and conquer Israel a millennium and a millennium and a half later - Babylon, Persia, Assyria - did so in order to increase their own empires, and not out of pure hatred of Israel, for hatred's sake.
Amalek, by contrast, had not even an unjustified excuse. He attacked us in the desert, near Egypt. He faced no threat - not even an imaginary threat - to his nation or his territory. Even the most paranoid Amalekite could not make the case for the attack. His attack on us was pure hatred for Israel and - more importantly - hatred for the holiness that Israel represents. Amalek saw the encounter with us in the desert as a chance occurrence - and responded to that happenstance with a murderous and unprovoked attack. For that attack, Amalek has to be exterminated.
Who was Amalek? His father was Eliphaz, who was a son of Esau and Adah, the daughter of Elon the Hittite (Genesis 36); the Hittite being descended from Heth, the son of Canaan, the son of Ham, the son of Noah (Genesis 10:1,6,15). Amalek's mother was Timna, Eliphaz's concubine (ibid. 36:12), the daughter of Sair the Horite (ibid. verses 20-22). Amalek is closely related to us - more closely even than Ishmael is: two of Amalek's great-grandparents were Isaac and Rebbecca.
Amalek (who was in the same generation as Jocheved, the mother of Moshe) has a half-brother called Reuel, son of Esau and Basemath, daughter of Ishmael (ibid. verses 2-4). And Reuel was another name for Yitro (Jethro), (the former) priest of Midian and father-in-law of Moshe (see Exodus 2:18, 3:1, 18:1, N
Amalek saw the encounter with us in the desert as a chance occurrence - and reacted with a murderous attack.
umbers 10:29; Mekhilta; Sifri, B'ha'alotkha 78; Yalkut Shimoni, Exodus 169).

Amalek saw the encounter with us in the desert as a chance occurrence - and reacted with a murderous attack.

Were these two the same man, or were they two separate men who happened to have the same name? Neither the Torah nor the midrashim answer this explicitly. However, the Torah does tell us that the sons of Reuel, the half-brother of Amalek, migrated to mount Sair (Genesis 36:9), and from there to the land of Edom (verse 17) - an eastward migration, the next stage of which would logically be Midian, the land of Yitro. Intriguingly, the Targum Yonatan (Genesis 36:12) records that Eliphaz, the father of Amalek, was the same Eliphaz the Temanite who was one of Job's three friends. For sure, Job lived in the east (Job 1:3), and his other two friends came from Shuah and Naama (ibid. 2:11), the general vicinity of Midian, a couple of hundred miles east of the Dead Sea.
This gives us a fascinating perspective on Amalek. Yitro - Reuel- Amalek's uncle "heard... of all that God had done to Moshe and to Israel his nation... and Yitro rejoiced at all the good that HaShem had done to Israel in rescuing them from Egypt." (Exodus 18:1,9) In stark contrast, Job's three friends - including Eliphaz, Amalek's father, "Heard of all the evil that had befallen him." (Job 2:11) The uncle and the nephew each hears what he wants to hear, views the world according to his lights, and reacts accordingly.
Amalek is the epitome of all that aspires to evil, just as Israel is the epitome of the good that God desires for the world. The two nations must inevitably fight one another to utter annihilation. It is the eternal war of good versus evil, which began that bright spring day in the desert 3,319 years ago, abated for a few decades and resumed when we entered the Land of Israel. It continued intermittently, peaking at the reign of King Saul 2,887 years ago.
Amalek's descendant, Haman, made another attempt to exterminate us all on Purim, 2,362 years ago. And in the intervening millennium, Amalek's descendants and ideological heirs have never ceased in their efforts to exterminate us. It is a war that will culminate when Mashiach comes, in the days "when HaShem your God will grant you respite from all your surrounding enemies." As the Targum Yonatan renders: "Eliminate the memory of Amalek from under the heavens, and even in the days of King Messiah, you shall not forget."