The Nazi party ruled Nazi Germany, which was one of the sides in World War II, opposing the United States Army and the Soviet Union. In 1944 they obtained a meteor from Thanagar, only to lose it in a clash with Vandal Savage and Rip Hunter's team.  In a version of 1942 that appeared after the Legends destroyed the Time Masters, causing aberrations throughout the timeline, the Nazis, allied with Damien Darhk, would destroy New York City with a nuclear bomb, prolonging World War II to the year 1947. This reality was averted by the Legends.
At one point, Adolf Hitler led the Nazi party, and started a bid for conquest. However, according to Rip Hunter, he was a "rank amateur" compared to the likes of Vandal Savage, as far as conquerors went.
According to Dr. Valentina Vostok, when the Nazis invaded the Soviet Union, the country was reduced to ashes, only to rise again stronger. This historical experience motivated Dr. Vostok in her desire to create a Soviet Firestorm.
The Nazis were at war with the United States in 1944. At Saint-Lô, France, the two sides were at the midst of battle, when the time-traveling Vandal Savage has arrived to claim his prize, a meteor from Thanagar, a source of great powers. Sergeant Rock, one of the American combatants, wondered whether Kendra Saunders, a recent escapee of Savage, was running away from the Nazis, and from whom else, if not them.
A military caravan, led by a Nazi officer, took hold of the Thanagarian meteor and was transporting it near Saint-Lô when it was confronted by Vandal Savage. Speaking to the officer in German, Vandal Savage demanded for the meteor to be handed over to him. The officer did not take the immortal's demands seriously, ordering him to move, only to be attacked by Vandal. The struggle was soon joined by the members of Rip Hunter's team, who, fighting both the Nazis and Vandal Savage, tried to make sure the meteor does not end in Savage's hands, while his prisoners, Kendra Saunders and Scythian Torvil, escape. The battle ended with all the Nazis present dead or otherwise knocked out.
Due to the timeline aberrations that started to appear throughout history after the fall of the Time Masters, there was a possibility for the reality, in which the Nazis attacked New York City, United States with a nuclear missile in 1942, to cement in time (in this reality, the Nazis would still lose World War II by 1947, but with even more casualties on all sides). Adolf Hitler's plan for this operation included the abduction of Albert Einstein and Mileva Maric and the cooperation of Damien Darhk, who provided the Nazis with uranium. A number of Nazi secret agents were sent to a Physics symposium, where they prepared to abduct Dr. Einstein, disguised as decorated members of the military, whose exquisite decorations were instantly recognized by Rip Hunter and Mick Rory as fake. After realizing their cover was blown, the Nazis opened fire at the symposium, before being killed by Rip Hunter. In the meantime, Mick Rory and Martin Stein conducted a "friendly" abduction of their own, taking Einstein to the Waverider. The other Nazis, spied on by Sara Lance and Ray Palmer, were conducting a deal with Damien Darhk, who later acted as the commander of a Nazi U-boat located in the waters near New York.
The Legends countered the Nazis' plan and went on to stop a Nazi U-boat under command of Damien Darhk from destroying the city. In his anger, Darhk ordered a nuclear attack on the Waverider, which was hit by the missile, but not before scattering its crew throughout the timeline, with Mick Rory left on the drifting timeship underwater, placed in time stasis. After being rescued by Oliver Queen and Nate Heywood in 2016, Mick used the time ship to gather the Legends once again, and finely safeguard both Albert Einstein and Mileva Maric, disrupting the Nazis' plans. Meanwhile, on the U-boat, a Nazi naval officer came into conflict with Damien Darhk over the disruption of their plans. After the Nazi suggested using another bomb, Damien denied, double-crossing the organization, and saying that his partner wants to "move on to other things". His partner being Eobard Thawne, the Reverse-Flash, all of Nazis aboard the U-boat were immediately slaughtered by the meta-human speedster's devastating attack.
The Legends would find themselves fighting the Nazis again very soon. After the time-traveling Eobard Thawne whisked the SS-Standartenführer Baron Krieger away from a group of Nazis in Paris, France, he presented him with a bargaining chip, a biomolecular enhancer that allowed to transform normal humans into meta-human berserkers for some time, in exchange asking for an item of some sort that Adolf Hitler didn't want to part with. A group of Nazis and Wehrmacht officers were stationed in a Paris club Folies-Bergére, amongst them General Gord von Runstedt, who asked Martin Stein, disguised as Max Lorenz, to perform for the group. Von Runstedt also uncovered Ray Palmer's identity, seeing him refusing to do the Nazi salute, leading to fight with the Legends and the JSA. Baron Krieger later used the biomolecular enhancer, transforming into a creature he dubbed "der Übermensch", to combat members of the Legends and the Justice Society of America that targeted the Nazi caravan transporting Askaran Amulet, with Krieger's forces capturing Ray Palmer and Amaya Jiwe. 
To buy some time, Palmer tried to banter with Baron Krieger, who threatened Amaya and forced Ray to produce more of Eobard Thawne's biomolecular enhancer. After Krieger left and a couple of Nazi-aligned soldiers entered, Ray and Amaya fought them off, freeing themselves. A number of Nazis followed them, along with the transformed Übermensch. The JSA and the Legends battled the Nazis again, with a bombing run finally destroying Baron Krieger. 
DC's Legends of Tomorrow
Behind the scenes
- Historically, Nazi is short for "National-Socialist", a word that described the members of NSDAP (National Socialist German Workers Party) – a party that took control over Germany in 1933, establishing a totalitarian regime under the leadership of Adolf Hitler. With the Nazi party at its head, Germany started World War II in Europe in 1939, only to lose it by 1945, with the Nazi Party branded as war criminals.
- In the real world, referring to the Nazis as a 'side' in World War II would be historically inaccurate; though the Nazis led Germany at the time, not all German combatants were members of the Nazi Party, and as such historians refer to the country itself as a 'side'. In DC's Legends of Tomorrow, however, the Nazis are always identified as the main opposing force during World War II, rather than "Germany" or "the Germans". This being said, Mick Rory has also referred to the Nazis as "Krauts", a derogatory term for German soldiers.
- In the DC Comics, rather than being an enemy of the Nazis, Vandal Savage claimed to be one of the founding members of the Nazi Party.