It's a large subject. The way they financed things changed repeatedly before and after the war, but I'll try to give a breakdown of the methods and some examples of each, focused on how these affected both the people of Germany and those outside it. If you are really interested I would recommend "The Wages of Destruction" which dives into the Nazi economy, and how it affected their decision making. Much of my answer will be sourced from it.

Their methods can be broken down generally into three major categories stealing, threatening, and selling.

Let's start with stealing, the one that people often think of when it comes to Nazi finance. Before the Nazi's would begin their war looting they committed large scale theft not only from the Jewish German population, but also from the rest of the german population.

Let's begin with the domestic theft. One example of this can be seen in Reichswerke Hermann Göring, a state owned steel conglomerate run by Göring. The conglomerate began with the theft of sites for iron ore mining from German steel industry in the Ruhr, moving private industry resources into the hands of the party/state. Another domestic example of theft was the looting of german museums of "degenerate art" (modern art) some of which was destroyed, but also much of which was sold on the foreign market to provided much needed foreign currency, which the state was almost constantly at risk of running out of. The final example I will give is the wide-scale theft from the German Jewish population, with the flight-tax forcing them to provide money if they wished to leave, and later with limitations on what they were allowed to own, the rest being stolen and used for financing (or simple personal looting by party members). This is certainly not a comprehensive list, but it does show that the Nazis were more than willing to steal public and private property from the German people for use in funding their war machine.

Let's move to the foreign theft. Wherever the Nazi's went plundering would occur. After Anschluss, the Austrian reserves were absorbed into the Reich's reserves, providing it a significant momentary access to foreign currency and bullion which allowed it to import more goods for war production. After the defeat of France, the french treasury was not ought-right annexed, but instead the French government was forced to pay exorbitant "occupation costs," which of course exceeded the true cost of occupation. These "costs" varied, but at one point was at least 20 million Reichmarks per day, enough to cover an occupation force well in excess of what was actually in France. In effect this was a looting of the French treasury. These monetary thefts were certainly part of financing the warmachine, but many other resources were also stollen. Large tracts of land were stollen with the goal of settling Germans there for a large scale agricultural colonization of Eastern Europe (universally by deporting or immediately killing the native population). Enormous amounts of the populations were also essentially kidnapped for wide-scale forced labor, with an estimated 10 million forced laborers in 1944.

Next we will move onto threatening. Domestically, the threat of violence was often used to force industry to follow along with the Nazi schemes. Industry leaders which did not play along, would be thrown in jail on trumped up charges or simply have their property stollen by the state. One example can be seen when the government planned to expand IG Farben's production of synthetic fuel, Goring demanded that the coal industry would provide resources to finance this expansion. As synthetic fuel was not a profitable enterprise it seems clear that this was simply a demand backed by a threat rather than a chance for coal industry investment in a new enterprise. Internationally the threat of invasion often loomed over Germany's trading partners. After the defeat of France, the USSR became one of Germany's largest trading partners and source of grain, at least partially to limit German desires to move west. Sweden continued to provide Germany iron ore it needed for steel production, a country that was certainly at risk of invasion. Other countries also provided resources in the hope of delaying or preventing invasion.

Finally, when bullying and attack were ineffective there was also actual trade. While Germany focused more and more of its domestic production on armaments, it also redirected the remaining resources towards the production of goods for export. Germany was continuously limited by natural resources in the form of oil, grain, iron ore, and other resources not abundant in German lands. It needed to import all of these for war production, which required foreign currency as its money was not worth much if anything internationally. For this reason it needed to produce exports with which to trade in its self created command barter economy. To Romania which provided it oil were sent weapons and planes (A trade which Hitler personally vetoed, and which Göring then personally forced through). To the USSR manufactured goods were sent for copper, manganese, and grain in the period immediately leading up to Barbarossa. These trades provided Germany with the resources to produce the weapons of war, but it also meant that the scant resources not destined for war still could not improve the lifestyle of the domestic German population, which desperately lacked for ways to spend the paychecks created by its "0% unemployment."

Of course there were other schemes and plans which I do not have the time to describe. If you are interested, I recommend reading MEFOs were another way to fund the war while keeping debts off the books.

I refer you to the thread Did Nazi Germany benefit or suffer economically from acquiring France and Poland? with thanks to /u/Estherke /u/Georgy_K_Zhukov and /u/commiespaceinvader