I'm sure this question has been asked before and there are better answers, but in a nutshell:

Prior to the end of World War II, Korea was a colony of Imperial Japan. After the end of the War, the Soviets supported the North and the US in the South; this was an agreement that came between the two, based in part on the way Berlin was partitioned up between them. The entire Korean Peninsula was devastated after World War II, and the Soviets spent a lot of money rebuilding Pyongyang and their industry, in particular. The Republic of Korea is established in the South, and the Democratic People's Republic of Korea in the North.

After a handful of failed skirmishes, the North invades the South on June 25, 1950, with Soviet aid. The UN, which the USSR is boycotting, is rallied into supporting an international alliance which lands in September and pushes back North Korea. When the UN forces continue North to the Yalu River, Maoist China becomes involved, which pushes the UN forces back.

There is a back-and-forth like this, ultimately ending with a militarized border at the Demarcation Line, pretty close to the 38th Parallel.

North Korea followed the path of the USSR and China and was heavily dependent on both for aid and support; this led to early industrialization and what seemed to be success, but they continued to default on loans and fell out of favor even among other Communist nations, especially after the collapse of the USSR, when they could no longer pivot between gaining assistance from either China or Russia and became more dependent on China.

The South, on the other hand, began with a military dictatorship and, at first, seemed more chaotic. Over time, however, South Korea received economic support from the US and turned toward a more democratic approach to governing. Victor Cha, in his book The Impossible State, describes today's free South Korea as the epitome of what we hoped to accomplish by supporting countries against Soviet aggression during the Cold War.

Sources:

The Impossible State, Victor Cha

An overview of North Korean history is also given in:

A Kim Jong-Il Production, by Paul C. Fischer

And these are two very, very powerful memoirs that give an inside account of how North Korea functions:

Dear Leader, by Jang Jin-Sung

The Aquariums of Pyongyang, by Kang Chol-hwan

I'm going to separate your question into three parts: why there are two countries? why North Korea seems so crazy? why South Korea is a democracy, today?

1. The simple answer is the Cold war. After WWII the United States and the USSR tried to claim as much Axis territory as possible to gain sway in the area and push their ideology of either Soviet Communism or U.S. Capitalism. Korea had been a territory of Japan since 1910 and thus the Americans moved into to liberate the country from the south and the Soviets from the north. Anyways the Korean war breaks out in 1950 which at first looked like either Korea could have control of the peninsula but after 4 years the war ended in a stalemate, a demilitarized zone separating them and a still separated Korea.
2. Even prior to the war, Soviet-like communism was being established in the north, aided by both the USSR and China. The issue was is the Koreans had gone from a brutal dictatorship under Japan that was largely Fascist with a deified leader. The Japenese wanted the Korean people to believe that they were an extension of the Japenese race and almost as pure as them. So many North Korea still held onto the idea that they needed a strongman that would be deified like the Japenese Emporer to protect themself from foreign invaders who might threaten the racial purity. Add on the fact that there were legitimate threats from South Korea, the U.S and much of the world of another invasion and their only basis of a communist government is the authoritarian USSR, thus North Korea installed, Kim Il-Sung as their leader. The Kim dynasty knew most of the world was/ is going to have a target on their back so to compensate for this and to create national unity the country turned to a "military first" policy painting that the country must have a strong military to resist the outside world.
3. Following the Koren war, South Korea installed a democracy but after much turmoil and constant rewrites of its constitution, the country had a military dictatorship of its own led by Park Chung-hee from 1963-1979. In the early beginnings of the country, it had struggled with famines and a poor economy and many turned to a dictatorship for answers. The dictatorship completely failed as it isolated the country as it shut down most foreign investments and foreign relationships in a coup, a new republic was formed. This new republic called the fifth republic which was still essentially a military regime with some democratic elements did make strides in the economy as they opened the country to investment and regained some diplomatic ties with Japan. In 1987, the government reformed to become the government it is today, with each person having western rights and being similar to most western democracies. The country today has become very similar to post-war japan as they move forward focusing on both tech and the automobile industry