When the Israelites entered the Promised Land the Amalekites had “tripped them up.” That is, they attacked the Israelites at their weak spot, or at their hindermost part or "tail," which was comprised of those who had lagged behind (Deut 25:17).
The Hebrew word for the hindermost part of the body is עָקֵב, which is used in the Hebrew Bible to refer to the rear echelon of an army (Genesis 49:19 and Joshua 8:13). That is, the rear echelon of an army is their hindermost part. The Amalekites attacked Israel at their hindermost part.
The exact same Hebrew word is also found in Genesis 3:15, where the serpent bites the "hindermost part" of the seed of the woman -- that is, the serpent attacks his heel, since the heel is physiologically the hindermost part of the human body. The seed of the serpent is therefore the enemy of God. It is the intent of the seed of the serpent “to trip up” the seed of God.
“Amalek” therefore is the enemy of the seed of God (Israel), who “trips up” at the most vulnerable time and opportunity. In the context of the Torah (first five books of the Hebrew Bible), the Amalekites in Canaan were not the seed of the Amorites, the Hittites, the Perizzites, the Canaanites, the Hivites or the Jebusites, but of Esau (Genesis 36:12). When Esau was born, his twin brother tried “to trip him up” at birth, and therefore the twin brother was named “Jacob.” Please note that “Jacob” (יַעֲקֹב) and “heel” (עָקֵב) are therefore related etymologically. Jacob later deceived his father Isaac in order to inherit the blessing of the firstborn -- that is, he had STRIVEN to inherit the promises made to Abraham and Isaac. Jacob “tripped up” his brother Esau to this end (of course he had already first secured the birthright from Esau in exchange for a mess of pottage, when Esau was most vulnerable because of hunger). Because Jacob was persistent and had striven with man (and now with God in Genesis 32:28) to inherit the promises to Abraham and Isaac, his name was changed from one who "trips up" to one who "persists" or strives with God (Israel).
The promised seed of Abraham therefore passes from Abraham to Isaac, and now to Jacob (Israel), whose sworn enemy in the Promised Land is Amalek. This hostility is not only between the biological descendants of Esau and Jacob, but also between those who are sons of promise to Abraham and those who reject the promise (like Esau).
Thus the Lord will have war with "Amalek" from generation to generation, notwithstanding that the biological Amalekites are no longer an identifiable race of people on the earth anymore.
Aside (not directly related to the question):
Let me illustrate from the New Testament. Jesus was born of Mary (seed of the woman). As the biological son of David through Mary, and therefore of Abraham, he was “tripped up” by his enemy. When Jesus quoted Psalm 41:9 (in John 13:18) he was referring to someone who would “trip him up.” (That is, he [Judas Iscariot] “lifting his heel [עָקֵב] against me”). Judas was his friend (in fact his disciple) and therefore his immediate access to Jesus was a point of vulnerability. Judas "tripped up" Jesus, since he was able to identify Jesus’ whereabouts when he was alone, and therefore betray him. In other words, Jesus (the seed) was "tripped up" by the seed of the serpent. Judas is the "son" of perdition (John 17:12). Jesus and Judas Iscariot were biological descendants of Jacob, but Judas Iscariot was not the son of the promise to Abraham but the son of perdition. Thus while there may no longer be any biological "Amalekites" on the earth anymore, Amalekites are also those who are enemies of the Promised Seed -- i.e., those who "trip up" the sons of Promise. Judas Iscariot was such a person.