The history of memorials and museums dealing with these periods of Berlin history is typically fraught with controversy, which is worth learning about: It tells much about the Berliner's soul.
The offices of the National Socialist's leaders and the headquarters of the Gestapo and Secret State Police once occupied a 15-acre tract in Mitte, the heart of Berlin. In these offices, genocide and terrorism were planned and administered. After the war, the complex was leveled, but when excavations uncovered the underground cells where prisoners were tortured, the site was preserved as the Topography of Terror, where visitors can look at the site and read and listen to its history. This, along with a visit to the Wannsee Villa, where Nazi leaders planned the Holocaust, can be a sobering introduction to an era.
The victims of the Holocaust and of the war that consumed all of Europe and America are commemorated by various sites in Berlin. The Memorial to the Murdered Jews of Europe, which opened 60 years after the war, both memorializes the individuals who died and invites visitors to literally walk into the experience of a regime gone wrong. The New Guard House, once a watchhouse used by Prussian troops, is now a memorial to all victims of twentieth-century wars. Its simple design -- a modern pieta by Kathe Kollwitz exposed to the elements -- is poignant and immediate. Finally, the Memorial to German Resistance is both a memorial to the five officers who were executed after a failed attempt to assassinate Hitler and an exhibition of the history of resistance.
Elsewhere in Berlin, it's worth taking a look at the monument at Humboldt University that recalls the 1933 Nazi book burning (Unter den Linden, next to Opera). Also remarkable is the German-Russian Museum-Karlshorst, the world's only joint memorial by former enemies to their shared history, in the building where the unconditional surrender by the Third Reich was signed in May 1945. (Zwieseler Straße 4/Ecke Rheinsteinstraße)
There are almost no traces left of the Berlin Wall, which divided East from West for 28 years. The Memorial to the Berlin Wall, however, documents the history of the Wall and the divided Germany. The House at Checkpoint Charlie -- which existed as a reconnoitering point for escape helpers during the Wall's existence -- shows many artifacts of the means East Berliners employed in their attempts to escape to the West.
A remaining segment of the original wall can be found near the Topography of Terror exhibit. Also, the famous, but crumbling, so-called East Side Gallery (along Spree River in Friedrichshain near Ostbahnhof) retains the characteristic original art that decorated the wall. These pieces were created in 1990 just after reunification, and have unfortunately suffered much deterioration.