Unlike its closest neighbors, China and Japan, Korea does not primarily use a form of Chinese characters for reading and writing. Instead, Korean is read using a unique alphabet that was created specifically for the language. This alphabet, called "Hangul," is extremely simple and it's possible to learn how to read it in only a few hours. Part of this is because the letters Hangul were designed both to be easy to read and to have their shapes match the sound they would represent.
There are 24 letters in Hangul, compared to the 46 used in the Latin alphabet (23 upper-case and 23 lower-case). Korean differs from many alphabets in that its letters are not written in a single horizontal string. The letters are always combined into blocks of syllables containing at least 2 letters but no more than 5. A perfect illustrated example of this comes from the Wikipedia article on Hangul: "although 한 may look like a single character, it is composed of three distinct letters: ㅎ h, ㅏ a, and ㄴ n." This can be further illustrated in this chart.
Prior to the invention of Hangul, the only way to read and write Korean was through the use of Chinese characters, the proliferation of which came from the introduction of Buddhist texts into Korea. Because of the complexity and large number of characters, for generations literacy was typically only a skill acquired by the wealthy with time and money for education. Hangul was developed in 1443 by Sejong the Great, king of the Korean peninsula, in order to bring literacy to the masses. It actually wasn't embraced immediately, it was banned for a number of years in the 1500s by kings that either felt the Chinese writing system to be superior or in order to subvert the uneducated. It wasn't until the Japanese occupation of Korea in the 1930s that Hangul became more standardized by the oppressed trying to maintain their national identity and eventually becoming fully adopted after the end of World War II.
It is important to note that Chinese characters are however still in use in modern Korea. They are taught in middle and high schools, used to differentiate between homonyms, used for names of both people and places and frequently used in newspapers.
Just for fun, here's McDonald's in Hangul.