Why is there a South Korea and North Korea?

Answer 1
The history of North Korea formally begins with the establishment of the Democratic People's Republic of Korea (DPRK) in 1948.

In the aftermath of the Japanese occupation of Korea which ended with Japan's defeat in World War II in 1945, Korea was divided at the 38th parallel in accordance with a United Nations arrangement, to be administered by the Soviet Union in the north and the United States in the south. The Soviets and Americans were unable to agree on the implementation of Joint Trusteeship over Korea. This led in 1948 to the establishment of separate governments, each claiming to be the legitimate government of all of Korea.

Answer 2
in short, north Korea which is contiguous to the land of china want a communist government while south Korean want a democratic government.

Answer 3
Stated simply, and in good grammar; When Japan left Korea after WWII, the monarchy was dissolved. New governments had to be formed, and with the US occupying South Korea, and Russia occupying North Korea, the influence of the occupying countries rubbed off of their occupants.

Answer 4
The short answer is because there was a shift in ideology, particularly in the northern part of Korea, after the end of WWII. The country started moving towards overt Communism, which the United States saw as a serious threat to US interests and security. The United Nations, with the USSR and China abstaining, voted to send in Allied troops to stop the spread of Communism. This was the Korean War. The war effectively ended in a draw, with a peace treaty being signed that separated Korea into North Korea (communist by name, although not following true communist principles any more) and South Korea (democracy).