Three mitzvot in the Torah relate to Amalek. The first is a positive commandment to remember what Amalek did to us, as the Torah says: "Remember what Amalek did to you on your journey, as you left Egypt" (Devarim 25:17). The second is a negative commandment not to forget what Amalek did to us, as the Torah says: "Do not forget" (ibid. 19). The third is a positive commandment to eradicate Amalek’s offspring from the world, as the Torah says: "Therefore, when the Lord your God grants you safety from all your enemies around you, in the land that the Lord your God is giving you as a hereditary portion, you shall blot out the memory of Amalek from under heaven" (ibid.).
Amalek symbolizes the root of evil in the world. Indeed, Amalek introduced the concept of Jew- hatred to the world. The Jewish people face a difficult struggle in this world. The idealistic, faith-based message that God assigned to Israel incites all the evildoers of the world to rise up and fight against the Jews. No other nation has been persecuted as the Jews have, from the destruction of the Temple, through the Crusades, the Inquisition, and the Khmelnytsky Massacres of 1648-9, and culminating in the terrible Holocaust that ravaged our nation. Amalek started it all.
Right after we left Egypt, even before we had a chance to coalesce and organize ourselves, Amalek attacked us, without any provocation or reason. Who did they attack? Slaves on their way to freedom after a prolonged period of servitude. Amalek is the nation whose very existence expresses hatred of Israel and, by extension, hatred of the Torah and the divine idea of repairing the world through kindness and truth. This is why the verse says, "‘Hand upon the throne (kes) of the Lord (Kah)!’ The Lord will be at war with Amalek throughout the ages" (Shemot 17:16). Rashi comments, noting that the verse uses shortened versions for two words – kes instead of kisei and Kah instead of the full Tetragrammaton – "The Holy One, blessed be He, swore that His name and His throne will not be complete until the name of Amalek is utterly blotted out."
A Jew is naturally kind and compassionate, and many mitzvot in the Torah further cultivate these traits within him. He would naturally be inclined to forgive Amalek. But the Torah commands us to remember what Amalek did and obliterate it. This way, we will remember that there is evil in the world, and that we must fight this evil to the bitter end, without compromise. Only then will we be able to perfect the world.