After World War II, Korea was temporarily divided between the US and USSR along the 38th parallel, with the USSR in control of the northern section and the US in control of the southern section. They originally agree to hold elections in 1948 to decide the government for the country, but the USSR refuses to hold them, so the north and the south remain separated. Because of the lengthened time that the two parts are separated, they begin to think of themselves as separate countries.
Once they get the "okay" from the USSR, the north invades the south in June of 1950. The United Nations responds, and the Security Council sent troops to defend South Korea (most of which came from the US). They were led by General MacArthur.
Once the fight gets too close to China's border, they enter the war and push the southern Korean armies back below the 38th parallel. Eventually they push back up to the 38th parallel again, right where they started. In 1953, everyone drops their weapons. The line at the 38th parallel becomes permanent to divide the two countries into North Korea and South Korea. It is called the DMZ, the demilitarized zone, since no forces at all are allowed on the exact line.
This war is called the Forgotten War. Although there were 54,000 deaths, it never got too much attention from the public.
This was important because it was considered another victory to the US - they had successfully stopped the spread of Communism from the USSR-controlled north into the US-controlled south. It was a perfect example of how a cold war works - the US was supporting South Korea and the USSR was supporting North Korea, but the two countries never directly fought against one another.