Why is the Korean war historically significant to the Cold War and to history overall?

The Korean War's main significance lies in the fact that it was the first time that the Cold War broke into actual military conflict. Prior to the outbreak of the Korean War, the great ideological struggle between East and West, between the forces of communism and capitalism, had taken place at the economic and diplomatic level. As both the United States and the Soviet Union were nuclear powers, any kind of direct armed conflict between the two would've been catastrophic, so both sides used proxy forces to help fulfill their strategic goals.
The Korean War would set the tone for the remainder of the Cold War. There would never be direct armed confrontation between the United States and the USSR, but there most certainly would be plenty of such conflicts between the respective opponents's proxies.