Looking at mythology, Korea is traced back to events which occurred at Baekdu Mountain (백두산, Baekdusan, 白頭山) around 2333 BCE in which a bear and a tiger wished to become human. They were given the opportunity on condition that they eat only garlic and mugwort and stay out of sunlight for 100 days. The tiger gave up but the bear succeeded, turned into a woman, and her descendants later became the Korean people and the first kingdom of Joseon. Mythology aside, this foundation story highlights the importance of Baekdu Mountain to Koreans, which is why it is often referenced in both Koreas and the Yanbian Korean autonomous prefecture in northeastern Jilin Province, China. As half of the dormant volcano lake is in China, and half is in North Korea, it is difficult to visit. To read more about other traditional myths, read the corresponding Wikipedia page as it is rather exhaustive.
There is archaeological evidence going back several thousand years of stone age farmers, a bronze age, and later iron so it is difficult to place a clear start date on when people first lived on this peninsula but it is clear peoples have lived here for at least several thousands of years. According to the Memorabilia of the Three Kingdoms (삼국유사, 三國遺事, samguk yusa) which a collection of historical accounts, legends, and folklore written around 1280 BCE, the previously mentioned legend is connected to the foundation of the Gojoseon kingdom (고조선, 古朝鮮). The Go at the front was added later as another Joseon kingdom also used the name from 1392 and something was needed to differentiate the two. It is also worth noting that National Foundation Day (개천절, 開天節, gaecheonjeol), which is a holiday in both modern Koreas, is held to commemorate the legendary formation of this first Korean state. The kingdom’s capital was in Liaoning City of modern day China, and later moved to Pyongyang. This kingdom saw the first rice cultivated, bronze and iron tools, and changes in pottery. When Gojoseon fell, it splintered into various smaller states which lasted for multiple decades.
When you learn about Korean history, most sources start with the Three Kingdoms of Korea which started around 57 BCE and ended around 668 CE but this disregards tribal states in the southern region and comparatively larger than tribal states more north around Manchuria. The big three that people focus on are Baekje (백제, 百濟), Silla (sometimes spelled Shilla, 신라, 新羅) and Goguryeo (고구려, 高句麗) which was later shortened to Goryeo (고려, 高麗). This last kingdom's shortened name is where the modern name Korea is derived and is the Korean spelling of Korea University.
Baekje was in the west area of modern South Korea and Silla to the east. The militaristic state of Goryeo took a significant amount of land extending from roughly where Seoul is to modern Russia and China and was largest in the 5th century. Baekje was known as a maritime power and is the reason many things were disseminated, from Buddhism throughout Asia, to culture and technology to ancient Japan. Under attacks from Goguryeo, the Baekje capital was relocated several times before their fall to Silla in 660.
Silla later took the northern kingdom of Goguryeo in 668 which unified the peninsula under one rule. It is debated when Silla established a centralized government and is either mentioned as the first of the three kingdoms to do so or the last. Regardless of their starting date, their diplomatic methods for alliances helped grow their nation. The last few hundred years of Silla (668–935) was a prosperous time for art, culture, and Buddhism.
Towards the end of Silla, the ancient states of Baekje and Goguryeo went through revival leading to Goryeo (고려, 高麗). Buddhism was the national religion and the Tripitaka Koreana was created on 81,258 wooden printing blocks at this time. International trade flourished with merchants as far away as the Middle East. Pottery and ceramics, especially celadon which were glazed a jade green color, greatly advanced. Metal-based movable type was created. Diplomatic relations extended through embassies into the southern kingdoms of China. The first Muslims arrived in 1024 and later built several mosques in the capital city of Gaeseong (개성시, 開城市). Soju was also first distilled during this kingdom and spread throughout the peninsula. The highest educational institution was Gukjagam (국자감, 國子監), which was the cornerstone of the Confucian reform leading up to an alter being built in the palace in 1398. The end of the dynasty saw a number of shipboard artillery advancements involving gunpowder artillery and perhaps an early version of the Hwacha; a rocket launcher that simultaneously fired up to several hundred arrows.
The final dynasty of Korea is that of Joseon (조선, 朝鮮). Founded by Yi Seong-gye in July 1392 in a coup d'état, the name of the kingdom was changed to Joseon, sometimes spelled Chosŏn or Chosun, to signify the change of leadership and quiet the threats of mutiny from nobles loyal to Goryeo. The change also resulted in the founding of a new capital in what is present day Seoul. Buddhism suffered a great loss going from the state religion to occasional persecutions while Neo-Confucianism became the new dynasty's state ideology.
King Sejong (세종, 世宗), the king you'll hear the most about, was the fourth ruler and one whose rule resulted in the title "The Great" (대왕, 大王, daewang) being added to his name. He encouraged advancements in scientific technology but is better known for personally creating and promulgating the Korean alphabet Hangul in 1443. Going from tens of thousands of Chinese-based characters to the phonetic alphabet shifted literacy from being reserved to the noble classes and extended it down to the common people. Viewing large scale literacy as a threat to the elite's power, Hangul's study and publication was banned in 1504 and it was hundreds of years later before it was widely used. Sejong is on the front of the modern 10,000 won bill and scientific advances are mentioned on the rear of the currency. A number of streets, parks, cultural buildings, and even the administrative city which holds a number of relocated government ministries, have been named after him.
The successors of Sejong undid a lot of his work and conducted a number of power struggles. Fighting wars with other nations only added to the instability leading the country to form an isolationist policy which is where the nickname "hermit kingdom" originates. Ascending the throne in 1724, Yeongjo (영조, 英祖) and his successor were known as having focused on bringing stability between factions and later decades saw increased isolation from the world stage. The last fifty years of the kingdom saw battles with at least two Asian nations and three Western nations and ended with the country being annexed in 1910 effectively ending the five centuries of Joseon rule. The end of World War II saw annexation end and the peninsula was split by yet other nations in 1945 leading to the Korean War starting in 1950. The war stopped in 1953 after prolonged stalemate but the war never formally ended. Foreign nations who militarily supported both sides in the armed conflict gradually left and both Koreas rebuilt.