There are basically two historical contexts going on here: the past and the present.
The past Wiesel's talking about is one of the darkest and most insanely horrific in history: the Holocaust. To try to sum that up in a short little intro paragraph would be ridiculous and reductive (you can't just cram a history lesson that big and important into a tiny space) so go here and get your knowledge on. Then come back.
When Wiesel's talking about the present, he's talking about the 1980s. But the hairspray and shoulder pads are miles away from where Wiesel's head is at: he's talking about the real-deal atrocities that have continued to plague Planet Earth even since the Holocaust ended.
World War II had been plagued with horrors, the Holocaust chief among them. But afterwards, with our role in the war done and over with and a post-war economic miracle pumping funds throughout the country, the United States declared the whole thing a success and settled into peacetime.
Of course, there was one problem (as there always seems to be). With the enemy-of-my-enemy relationship the U.S. enjoyed with the USSR during WWII over, tensions smoldered straight into the Cold War. Russia and the U.S. never officially fought each other on the battlefield, but the entire 40+ year span of the conflict was filled with proxy wars, with the U.S. trying to block anything that whiffed of communism at all costs. In doing so, we became uncomfortable bedfellows with a lot of unsavory terrorist groups and tyrannical dictators…because at least they hated commies as much as we did.
Peacetime, it would seem, hadn't been so peaceful after all. And with Reagan's administration, Cold War tensions were back to full boil.
To make matters worse, in 1986 South Africa, still held by the Dutch, was under the reign of Apartheid: a government where African citizens (you know, the people who were actually from there) were held as second-class citizens. Many western nations were flexing their diplomatic muscles against South Africa, but Reagan's America was oddly silent. Activists and public figures like Wiesel urged the United States to break the silence and throw their weight in the ring.
Three years after World War II, the United States, alongside Britain, cut the nation of Israel out of Britain's holdings in the region of Palestine, allowing unlimited Jewish refugees to move into their ancestral homeland. As powerful of a gesture as it was, the young nation of Israel was embroiled in conflict from the very start. The Palestinians of, well, Palestine were none too happy about the British carving out a chunk of their current country of residence, and tensions between Israel and Palestine rage on to this day. Four years before Elie Wiesel delivered his speech, Israel had invaded Lebanon to try to squash the Palestine Liberation Organization operating out of there.
The Cold War still lay on top of all of global politics like a shroud with dumbbells stitched on the edges. America and the Soviet Union had been throwing down in the Middle East since the Soviet Union invaded Afghanistan in 1979, and the United States, in their efforts to beat the Ruskies, funded groups such as the Taliban, setting the stage for our involvement in the region today.
Oh, and about the Cold War: both sides still had the atom bomb. Hanging over all of these global conflicts was the threat of mutually assured destruction. As if the Holocaust wasn't bad enough; mankind could have very well had a Nuclear Holocaust on their hands if cooler heads hadn't prevailed.
Elie Wiesel received the Nobel Peace Prize in 1986 because he had spent his entire life after the Holocaust advocating for the sanctity of human rights and the need for world peace. And in 1986, peace was definitely something the world could use a lot of.
Huh. Guess 1986 wasn't so different from now, after all.