The mitzva to destroy Amalek is primarily incumbent upon the people of Israel as a whole. Indeed, the Sages taught that the people of Israel were commanded to fulfill three mitzvot upon entering Eretz Yisrael: first, to appoint a king; then to wipe out the descendants of Amalek; and then to build the Holy Temple (San. 20b).
Indeed, after the people of Israel became coalesced as a nation in its land, appointed a king in Shaul, and his kingdom stabilized, the prophet Shmuel approached him, saying:
I am the one the Lord sent to anoint you king over His people Israel. Therefore, listen to the Lord’s command! “Thus said the Lord of Hosts: I am exacting the penalty for what Amalek did to Israel, for the assault he made upon them on the road, on their way up from Egypt.” Now go, attack Amalek, and proscribe all that belongs to him. Spare no one, but kill alike men and women, infants and sucklings, oxen and sheep, camels and asses!” (1 Shmuel 15:1-3)
However, King Shaul did not fulfill the mitzva properly. He had pity on King Agag of Amalek and on the best of the sheep and cattle. As a result, God took the kingdom away from Shaul and gave it to David. Nevertheless, the damage that was already done was devastating. Because of Shaul’s weakness and compassion, many Amalekites survived and continued harassing Israel. A few years later, a band of Amalekites attacked Tziklag, where the families of David and his men lived, burning down the city and taking all the women and children captive. With God’s help, David and his men managed to rescue the captives and vanquish the marauders. But since David was not yet king and did not have the army of Israel at his disposal, he was unable to eradicate them. According to 1 Samuel 30:17, “four hundred young men…mounted camels and got away.” Apparently, other groups of Amalekites survived elsewhere as well. Even after David became king and continued to fight against the Amalekites, he was unable to destroy them all, because they were scattered throughout the land. The Sages also relate that because Shaul delayed killing Agag, Agag’s line was preserved, eventually resulting in the birth of Haman the Agagite, who attempted to wipe out the Jewish people (Megilla 13a).
Even though the mitzva to eradicate Amalek is primarily incumbent upon the Jewish people as a whole, every individual Jew is commanded to fulfill it, as well. Therefore, if a Jew meets an Amalekite, and has the ability to kill him, but refrains from doing so, he has neglected this mitzva (Sefer Ha-ĥinukh §604). The descendants of Amalek are currently unknown, but if one would ascertain that a particular person is an Amalekite, and that person follows their ways, it would be a mitzva to kill him.
I asserted that some Amalekites remained after Shaul neglected to kill Agag. This is clear from bb 21a-b, which relates that David’s general Yoav erred in killing only the men of Amalek but not the women, because his school teacher had taught him an incorrect interpretation of the relevant verses. This incident took place during the war mentioned in 1 Melakhim 11:15-16, which was fought mainly against Edom, i.e., the descendants of Esav. Since the Amalekites represented only a small part of the descendants of Esav, this implies that there were groups of Amalekites within the Edomites, and it was concerning those Amalekites that Yoav erred. This explanation is cited in Responsa Kol Mevaser 2:42.