Regarding the settlement of the issue of the ownership of the Oregon Territory, this territory was jointly owned by the United States and Great Britain for some time. As the years went by, there came to be more conflict between the two countries as to how much of the territory would belong to each country. This conflict led to demands among some Americans for "Fifty-Four Forty or Fight." This was a demand that all of the Oregon Territory up to the Fifty-Four Forty line should be American territory.
Eventually, the issue was settled peacefully. The US took the territory up to the 49th parallel, where the border between the US and Canada is today. As the almanac link below says,
After lengthy though peaceful negotiations with the British, the United States obtained on June 10, 1846, all the territory south of the 49th parallel (the line of latitude that now defines the United States's northern border with Canada).