Nazi

The Nazis, or National Socialists, were a racist, fascist, and expansionist political party in Germany. They came to power under Adolf Hitler in 1933, and launched World War II later in that decade.

Pre-war
The Nazis were influenced by first the Timewyrm, who took up residence in Adolf Hitler's mind, and then the War Chief and the War Lords, who detected the Timewyrm's psychic powers within him and hoped to use him to conquer the universe. The Seventh Doctor, by becoming a personal advisor to Hitler, was able to first expose the machinations of the War Lords and later expel the Timewyrm (which left Hitler disoriented enough to allow the Dunkirk evacuation). (NA: Timewyrm: Exodus)

In 1936, Nazi sympathizers, including King Edward VIII and Sir Oswald Mosley, attempted a coup to place a Nazi-friendly government in charge of Great Britain, but were foiled by the Sixth Doctor and Peri Brown. (PDA: Players)

That same year, the Seventh Doctor and Melanie Bush met a Nazi scientist named Emil Hartung, and accidentally gave him ideas that would lead to the development of a stealth bomber. (NA: Just War)

World War II
In 1938, the Nemesis statue influenced Hitler to annex Austria. (DW: Silver Nemesis) This event was later considered by some to be the start of the War, although open hostilities did not begin until the invasion of Poland in 1939.

In 1940, the German Air Force began an air campaign known as the Battle of Britain. The campaign was designed to establish air superiority over the British and destroy their defenses.

By 1941, the Nazis were able to launch a scaled-back version of Operation Sealion and conquer the Channel Islands. From there, they intended to use Hartung's weapon to force an Allied surrender. Although accounts differ, either Chris Cwej and Roz Forrester (NA: Just War) or the Seventh Doctor and Jason Kane (BFA: Just War) managed to destroy the bomber.

Having failed to establish air superiority, the Nazis turned to a terror bombing campaign known as the Blitz. The Doctor and his companions were involved in the Blitz a number of times, although rarely engaging the Nazis any more directly than through their bombs. (DW: The Empty Child, TW: Captain Jack Harkness, DW: Victory of the Daleks) In one instance, an attempted invasion by the Cybermen, using the Blitz as a cover, led the Seventh Doctor, Ace, and an American private detective to expose secret Nazi operatives within Britain. (PDA: Illegal Alien)

In 1941, a group of Nazis led by Koenig attempted to use a device which would block British radio signals, allowing German ships to attack without being detected. The plan was foiled by George Woods and the time-travelling Clyde Langer. (SJA: Lost in Time)

During the height of the War, both the Sixth Doctor with Brigadier Lethbridge-Stewart (PDA: The Shadow in the Glass) and the Seventh Doctor with Ace (NA: Timewyrm: Exodus) met repeatedly with Hitler and other high Nazi officials.

In 1944, the Seventh Doctor and Ace arrived at Colditz Castle, a Nazi POW camp, and accidentally left behind a laser, which allowed the Nazis to turn defeat into victory and conquer most of the world. (BFA: Colditz) In this alternate timeline, the Nazis captured the TARDIS. Elizabeth Klein, an ardent Nazi, brought the TARDIS back to 1944 and inadvertently allowed the Eighth Doctor to restore history, stranding her in the prime reality. (BFA: Klein's Story) Elizabeth Klein subsequently became a companion to the Seventh Doctor, but plotted to restore her original timeline. The final outcome of a series of manipulations led to Elizabeth Klein the Nazi being replaced by Elizabeth Klein the UNIT hero. (BFA: A Thousand Tiny Wings, Survival of the Fittest, The Architects of History)

In 1945, Hitler committed suicide, and the Nazis surrendered shortly thereafter. (PDA: The Shadow in the Glass)

In an alternate timeline, the Nazis were totally victorious. They dominated their world, all enemies had been "Sent East", and Adolf Hitler and Eva Braun had conceived a child, August. This world was visited by agents from the Great Houses, namely a Cwej-Prime in the service of House Mirraflex. This Cwej-Prime adopted the name "Herr Abschrift" (Mister Copy, a play on the fact he is a duplicate of Chris Cwej) and offered the Nazis of "Germany Alpha" the ability to not only travel to other worlds in their galaxy, but also other worlds across the multiverse. This led to the creation of a Reich that reached across hundreds, if not thousands, of worlds. It was only defeated by forces unified by a Roman Empire that (like the Nazi Reich) had never fallen to their enemies, with the support of Faction Paradox, possible covert agents of the Enemy, and perhaps even the Doctor. (FP: Warlords of Utopia)

Neo-Nazis
After the allied victory in World War II, the Nazi Party was disestablished and banned in occupied Germany. However, neo-Nazi movements ranging from organized political parties to skinhead gangs continued to cause trouble for decades to come.

In 1988, a neo-Nazi group led by De Flores attempted to take control of the Nemesis statue and use it to establish a Fourth Reich, but they were foiled by the Seventh Doctor and Ace. (DW: Silver Nemesis)

In 2001, another neo-Nazi organization, led by Adolf Hitler, Jr., pretending to be his father preserved by stolen Vvormak technology, planned to conquer the world with the help of demonic imps, but the Sixth Doctor and the Brigadier managed to foil the plans. (PDA: The Shadow in the Glass)

In 2593, on Tyler's Folly, Bernice Summerfield encountered the SSSSSSS ('Stella Stora Sigma Schutz-Staffel SturmSoldaten') which was considered the most ineffective Nazi group since the Outer Hebriden National Party. (BNA: Down)

Behind the scenes

 * Terry Nation modeled the Daleks in part on the Nazis, and in particular drew inspiration from them in developing Kaled society for DW: Genesis of the Daleks.
 * See here for more on the Nazi/Dalek similarities