There is a positive mitzvah to remember how Amalek cravenly ambushed the weakest members of the Jews who left Egypt (Mitzvah #603). Here, we have a negative mitzvah prohibiting us from forgetting that which Amalek did to our ancestors. Not only are we not to forget it, we also are not to forgive it; that would cause us to come to embrace the Amalekite nation and would effectively nullify the mitzvah.
The reason for this mitzvah, as we said regarding Mitzvah #603, is to remember God's providence over Israel and how those who attack Him meet their downfall.
This mitzvah applies in all times and places. It is discussed in the Talmud in tractate Megillah (19a-b and 29a-b) and is codified in the Shulchan Aruch in Orach Chaim 685. This mitzvah is #59 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos and #194 of the 194 negative mitzvos that can be observed today as listed in the Chofetz Chaim’s Sefer HaMitzvos HaKatzar.