🔼The name Amalekite in the Bible
The people's name Amalekite is initially mentioned in the story of the War of Four Against Five Kings (Genesis 14:7). Later the name Amalek becomes the name of a grandson of Esau (Genesis 36:12) who becomes a chief of Edom (v16). In his fourth sermon, the prophet Balaam, declares Amalek the founder of the "first nation," although it isn't clear if the son of Esau is meant, or rather the founder of the original Amalekites who lived in Canaan at the time of Abraham.
The Amalekites that feature prominently in the rest of the Bible are probably not the descendants of Esau, simply because that's never mentioned, and the descendants of Esau are collectively referred to as Edom and Edomites.
The Amalekites stage an attack on Israel at Rephidim, just after their departure from Egypt (Exodus 17:8-16). Moses and Joshua defeat them and God tells Moses to make a note that He will utterly blot out the memory of Amalek from under heaven (Exodus 17:14), obviously not meaning the name Amalek or else we wouldn't know about him, but Amalek's participation and role on the world stage. The Amalekites bounce back and try again during the days of the judges.
Under Eglon, king of Moab, Amalek successfully attacks and occupies Israel for eighteen years until judge Ehud, son of Gera, valiantly assassinates king Eglon and Israel kicks out Amalek and Moab (Judges 3:12-30). Later the Amalekites come back (Judges 6:3) but are defeated by judge Gideon. Israel becomes a monarchy under Saul, who proceeds to wage war on pretty much everybody, including the Amalekites (1 Samuel 14:48). God commands Saul to finally destroy Amalek entirely (15:2-3), but Saul has other plans, lets Amalek's king Agag and all the livestock live and sets up a monument to celebrate his own heroism (v12). God regrets having made Saul king and Saul's decline sets in (15:11).
Before David becomes king, the Amalekites destroy his city Ziklag and abduct the inhabitants, including David's wives Ahinoam and Abigail (30:5). David and six hundred of his men pursue the Amalekites, slaughter almost all of them (30:17) and retrieve the prisoners. Whatever Amalekite remnant remains is wholly obliterated by a band of Simeonites during the time of king Hezekiah (1 Chronicles 4:43).
🔼Etymology and meaning of the name Amalekite
See for a discussion on the possible meaning of the name Amalekite our article on the name Amalek.