The allied forces of the British and American armies put up stations in Iceland during the second World War. For example, they put up airports in Reykjavik and the Reykjanes Peninsula, one of which you may have landed on to get to Iceland. The Allied Forces used their bases in Iceland to thwart German naval ships and submarines and protect their shipping routes in the Northeast Atlantic.
Close to Reykjavik they set up bases in Hvalfjordur (Whale Fjord) during WWII where their fleets, both naval and commercial, paused on their routes and gathered forces for their journeys. Large fleets often lay anchored right outside Hvammsvik containing some of the largest war vessels of the world.
On Hvítanes Peninsula, one of the more picturesque waterfront trails in Whale Fjord, the British army built a naval station to protect the fjord and its fleets. On the Northern side of the fjord the US Army built a large fuel station that made sure that the vessels that stopped by in Hvalfjordur fjord had plenty of oil for their journeys.
Winston Churchill once came to Iceland during the war, on a ship, and that ship came here to Hvítanes, before Churchill went on to meet Icelandic ministers in Reykjavik. Special steps were built for him to mount up the bank, rumors have it, but it is unsure if he actually stepped foot on them ;)