Greenland was not a major theatre of operations during World War II, to say the least. However, like every other corner of the world, it was touched by the conflict. Greenland was a strategic location for both the Axis and the Allies because weather that developed over Greenland would soon be affecting The North Atlantic and Northern Europe. In a war where information was key, knowledge of the conditions that bombing missions and U-Boat raids would take place under was vital.
Greenland was also important because it was the colony of a European country (Denmark) that was overrun by the Nazis. A month after the invasion of Denmark, the United States was invited by the Danish ambassador into Greenland to protect the territory. This was perhaps an intentional challenge or testing of limits by FDR, 18 months before Pearl Harbor he was intentionally putting the United States in direct territorial conflict with Germany.
In 1941, several expeditions were sent to Greenland to plan for possible naval bases and air bases. Soon after this, the United States instituted a naval patrol of Greenland. This patrol was not actually carried out by theUnited States Navy, but by the United States Coast Guard, perhaps because Greenland could be considered roughly an extension of the United States coastal waters. In September of 1941, the Coast Guard seized a Norwegian trawler sent to establish a German weather station on the East Coast of Greenland. This was the first time the United States had directly come into contact with Germany.
After the declaration of war, Greenland became important as a "unsinkable aircraft carrier" on planes making ferry flights from the United States to England. Several large airfields were built, which by itself was a great work, since constructing a modern airfield in a country with no supplies and rather inclement weather was quite a feat of engineering. but like every other American engineering project in World War II, it was somehow managed.
In addition to this, several more attempts were made by the Germans to land observation teams on Greenland, as late as 1944. However, they were all roundly rebuffed. After this point, the rapid dissolution of the Nazi empire would mean the Germans had more immediate things in their plans then trying to send weather teams to Greenland.
Like every other nation, the war markedly changed everything about Greenland. The influx of American technology and supplies would mark the introduction of Greenland into the modern world. Several of the air bases that were constructed during World War II served as bases during the Cold War and now serve as civilian airports.