The History of Korean language can be divided into roughly 4 stages: Ancient (1st millenium), Medieval (~19C), Early Modern (late 19C~early 20C) and Modern (after 1945). Korean alphabet was invented in 1444; the spoken vernacular wasn't scribed and codified before. There were some efforts to create a writing system for Korean using Chinese characters, but they were too complex and unsuccessful.
'Ancient' period refers to the era of the Three Kingdoms of Korea. Korean peninsula and southern Manchuria was divided by many tribal states in early half of the 1st millenium AD. Among these, Buyeo, Okjeo, Dongye, and Goguryo formed a distinct linguistic group and Jinhan, Mahan, Byeonhan formed another. Among these, the three most important states- Goguryeo, Baekje(Mahan), Silla(Jinhan) are called the 'Three Kingdoms.'
Whether the Three Kingdoms of Korea spoke the same language or not is still a subject of controversy, since only a few hundred words are reconstructed at this point. By the time of Goryeo dynasty the peninsula did speak more or less same language, but whether Goryeo was based on Goguryeo or Silla variety is another issue of controversy. Generally South Korean linguists consider Medieval Korean to be based on Sillan, and the North Korean linguists consider it to be Goguryeo-based.
Vernacular language was scribed only after the creation of Hangul in 1444, and it slowly gained its status as literary language ever since. The written vernacular belonged to the commons and women; the language of education, administration, and academics was still Chinese. This Medieval Korean differs a lot from the Modern Korean, both in terms of grammar and vocabulary. Medieval Korean had three extra consonants and one extra vowel. It can be compared as Chaucer to the Modern English; moderns can't read it without special education. A variety of Medieval Korean is still spoken in Jeju Island, where its language remained relatively unchanged from 15C Korean. It's mutually unintelligible with the mainland dialects, but it's labeled as a 'dialect' because of political reasons.
As the Confucian Fundamentalist Joseon Dynasty fell, the usage of Chinese as literary langauge significantly decreased, and Korean gained its status as nationl symbol again. In late 19C a new wave of literature called 'New novella' spreaded throughout the country, and some of them had eduational or political motives. The demand for the standardization increased as Korean became a major literary language again, and the first "standard" Korean appeared in 1930s, despite the efforts of Japanese Colonial Government to sabotage it. The language of this period is 'understandable' for most modern Korean population, but most of them will find it somehow 'different.'
Modern Korean starts after the liberation and Korean War. Two 'standard' forms appeared as the country divided into half; "Standard Language" of South Korea and "Cultural Language" of North Korea. Both standard languages were primarily based on 1930s Korean Institute standardization, and they remained relatively unchanged. But the new influx of foreign loanwords from English (in South Korea) and Russian (in North Korea) and different idiomatic usages make clear distinction between 'Modern' and 'Early Modern' Korean.
Any comments/questions are welcome.