The Korean War really started in 1945 when Japanese rule in Korea ended and the Allies set up both the Democratic People's Republic of Korea and the Republic of Korea (North and South Korea respectively). The Soviets immediately set about building up North Korea, they sent supplies, industrial,technical experts, etc... The Americans didn't help South Korea as much. Both countries had dictators,the North had Kim-Il-Sung and the South had Syngman Rhee. In these early years and the years following the Korean war the North was actually better off economically than the South (that changed around the 70's).

Kim-Il-Sung along with Russian advisers began to plan for a war against the South. The war was declared in 1950 and looked to be a quick one. The South Korean army was pitiful compared to the North Korean army and the few US troops in Korea couldn't do anything to stop the North Koreans. The North pushed the South all the way back to the port of Pusan. The UN security council put forth a motion to intervene to help the South. Usually, the Soviets would have vetoed the motion, but they were currently boycotting the UN and thus couldn't veto the motion. With UN reinforcements the South pushed the North all the way back to near the Chinese border; the threat of an American ally on their border caused anxiety for the Chinese, so they decided to intervene and pushed the UN back to the original border between South and North Korea. After this the war stalemated. The US began to bomb North Korea (dropping more bombs than they did in WWII). Hostilities ceased in 1953.

Sources:

The Impossible State: Victor Cha

The Korean War: William Stueck