Who were the Amalekites? What made them so bad?
The Amalekites were the descendents and followers of Amalek, grandson of Esau (Genesis 36:12,16), brother of Jacob also known as Israel. As such, the Amalekites weren't total foreigners to God. Esau was the one who had sold his birthright and his part in God's promise. He had been part of God's covenant people, but he valued his own apetites more. So the Edomites (Esau's descendents, including the Amalekites) were people who had opted out en masse of the covenant which defined God's people.
They weren't Canaanites. Israel was not a threat to them; Israel was not going to take their land. Their relationship to the Amalekites was like their relationship to the other Edomites when they said "Please let us pass through your country. We will not go through any field or vineyard, or drink water from any well. We will travel along the king's highway and not turn to the right or to the left until we have passed through your territory." (Numbers 20:17)
But the Amalekites really really didn't like Israel. At the very birth of the nation of Israel, when they came out of Egypt and were at their most vulnerable, before they even got to Sinai and when they didn't even have any water, the Amalekites came and attacked them (Exodus 17:8). Israel were forced to fight their very first battle, fighting for their lives against the Amalekites, under the leadership of Moses. After God gave Moses an amazing victory, Exodus says this:
Then the LORD said to Moses, "Write this on a scroll as something to be remembered and make sure that Joshua hears it, because I will completely blot out the memory of Amalek from under heaven."
Moses built an altar and called it The LORD is my Banner. He said, "For hands were lifted up to the throne of the LORD. The LORD will be at war against the Amalekites from generation to generation."
Exodus 17:14-16, NIV
The Amalekites were the people who hated Israel, right from the start. And though Moses said that God would be at war it looks very much as if it's the Amalekites who are at war with him. Israel have a lot of wars between Moses and Saul, but they never once attack the Amalekites.
The Amalekites attack Israel though. In Numbers 14:45, they attack Israel again while they are still in the desert. In Judges 3:13 they join in with the Moabites in attacking Israel. In Judges 6:3, they invade Israel "whenever the Israelites planted their crops", and together with the Midianites "did not spare a living thing for Israel, neither sheep nor cattle nor donkeys." Later in Judges 6 and 7 they invade again and are fought off by Gideon. The Amalekites show that generation after generation, they are at war with Israel and with God. They had attacked Israel at least five times over a period of about 250 years.
Even long after Saul (and Saul's successor David) have fought against and mostly eradicated the Amalekites, we get one more Amalekite coming up. 600 years after them, the Persians are ruling the whole area, and a man called Haman, an Agagite gets a lot of power. "Agagite" probably means that he was descended from the Amalekite kings, known as Agag.
After these events, King Xerxes (of Persia) honored Haman son of Hammedatha, the Agagite, elevating him and giving him a seat of honor higher than that of all the other nobles. All the royal officials at the king's gate knelt down and paid honor to Haman, for the king had commanded this concerning him. But Mordecai would not kneel down or pay him honor.
Then the royal officials at the king's gate asked Mordecai, "Why do you disobey the king's command?" Day after day they spoke to him but he refused to comply. Therefore they told Haman about it to see whether Mordecai's behavior would be tolerated, for he had told them he was a Jew.
When Haman saw that Mordecai would not kneel down or pay him honor, he was enraged. Yet having learned who Mordecai's people were, he scorned the idea of killing only Mordecai. Instead Haman looked for a way to destroy all Mordecai's people, the Jews, throughout the whole kingdom of Xerxes.
Esther 3:1-6, NIV
The Amalekites weren't just any old people. They were the nation who more than any other tried to destroy Israel. They had been trying to eradicate and plunder Israel from the very birth of Israel, 200-400 years before the command in 1 Samuel 15, and they would continue for another 600 years.
The Amalekites were vicious as well, and were noted for killing children (1 Sam 15:33).
That explains some of the background to the conflict in 1 Samuel 15. It shows that what is being commanded is an act of war in a conflict which the Israelites didn't start, and which was never going to be resolved by negotiation.