What is the significance or result of the German invasion of Denmark and Norway?
Norway was an important source of iron ore, but there were two more important reasons for the German invasion. Primary was the strategic location of the country, and if the British had seized Norway Germany would have been outflanked to the north before the war was well underway. This not only would have been disastrous in terms of losing control of the northern seas and the exit from the Baltic, but would have left the British in control of a direst route to Finland and Russia. The primary reason Hitler invaded Norway was to forestall this, which was indeed the plan of the British.
Although those two reasons were important, there was a third very important reason. That was the company Norsk Hydro, the world's only commercial source of deuterium, known as "heavy water." This is essentially water with an extra hydrogen atom, and which was used as the moderating agent to control the reaction of uranium in atomic research. The method of creating an atomic bomb which was being pursued by German scientists depended on heavy water.
German, Russian and British scientists had been working on an atomic bomb since before the First World War. Norsk Hydro's production was immensely increased during the war, but the shipment of large quantities was stopped by various means. There were several attempts at sabotage, some successful, and one major commando raid on the plant. It could not be successfully destroyed by bombing because of its position inside a steep valley at the base of a mountain. In February of 1943 the facility was heavily though temporarily damaged by a raid carried out by Norwegian and British commandos, some of whom had been living on the snow covered mountains for months. On 20 Feb., 1944, the Germans shipped a large quantity of deuterium in barrels on a ferry down Lake Tinnsjo; the ferry was sunk by a time bomb set by Knut Haukelid, a Norwegian commando. This was essentially the end of Germany's attempt to create an atomic bomb, although it was actually beyond their capabilities. They simply did not have the money or manpower available to carry out the massive industrial effort required at the time.
Norway was an important source of iron ore, from which steel is made. Hitler needed it for his war machine and he wanted to deny it to the British. Norway also had the world's only source of heavy water which was used at the time for nuclear research. By capturing it, Germany sprinted to the lead in the search for the atomic bomb.
Lastly, the German Kriegsmarine, or navy, took a serious hit in the invasion, losing ten destroyers and six U-Boats. This would make it more difficult for them to invade England later.
It was also quite impressive that Germany was able to conquer Norway in a month, since it's a nation that is one thousand miles long, in rugged mountains and fjords, with a third of the country above the Arctic Circle.
Hitler really needed Denmark to get at Norway, and to guard the entrance to the Baltic Sea. The country was taken in only four hours by German forces, although the resistance afterwards was very active.
In regards to Norway specifically, they were largely unprepared for the German invasion, and militarily it was not much of a contest. The British actually attempted a poorly planned and executed operation to oppose the German invasion but eventually withdrew after rather heavy losses. The German army quickly routed what little resistance there was and the King left for exile and ran his government from London while a puppet government was installed by the Nazis.
There was a prolonged and serious resistance movement that actually grew throughout the war. There were other programs including one in which many German soldiers fathered many children with Norwegian women who were shunned and ridiculed after the war. There were other long term effects, among them a prevalence towards stockpiling food and other supplies after a fear of the famine and scarcity of other goods during the occupation.
Of course, the main significance of this event was to the people who were affected by it -- especially Jews in Denmark.
In more overall terms, the main impact of this, in my mind, was economic. A major reason for this invasion was to take the port of Narvik in Norway. This would allow important shipments of iron ore from Sweden to get to Germany year-round (because Narvik is open all year where Sweden's northern ports are not).
So this invasion probably allowed Germany to be stronger than it would have been because it allowed them to get iron more easily than they would have otherwise.