Taegukgi 태극기 (Meaning: Taeguk flag)

The South Korean flag, also known as the Taegukgi. Consist of three different parts; the rectangular white background, the red and blue Taeguk placed in the center, and four different trigrams (selected from the original eight (Chinese: Bagua)) placed in each corner of the flag.

The background has a white colour, which is a Korean traditional colour. The colour white represents peace and purity.

Taeguk is the blue and red circle found in the center of the flag. The upper half circle, red, of Taeguk means yang and the lower half circle, blue, means yin. This symbol comes from the philosophy of yin and yang, and represents the balance of the universe.
Because the taeguk is associated with the national flag of South Korea, it is often used as a patriotic symbol, as are the colors red, blue and black.

The four trigrams together represents the principle of movement and harmony. Each trigram represent one of the four elements; earth, water, air and fire. Each trigram consist of three long and/or short black bars (called kwae). Each "kwae" consists of three bars, either broken or unbroken. A broken bar represents yin, while an unbroken bar represents yang.
It is said that these four trigrams move in an endless cycle, both back and forth,  in their pursuit of perfection.

Check the table below where the four trigrams are carefully described.

(Source)

 The Korean National Flag has a philosophical meaning. It originates from the old oriental philosophy known as the theory of Um-Yang, or in Chinese pronunciation Yin-Yang. Yin represents dark and cold things (like hell, water, death, evil and night.), while Yang, on the other hand, represents something bright and hot (like heaven, fire, life, good and day.). 

The idea of Yin-Yang is said to originate from the old Korean philosophy of Samshin, meaning three gods.

An old book called Chuyok or Iching in Chinese, written several thousands years ago, claims that all objects and events in the world are expressed by the movement of yin and yang. 

Taeguk is said to symbolize a liquid fluid, always changing and moving. This explains the wavy line between the two colours. The red part represents earth, while the blue part represents heaven. The whole idea of Taeguk is to show heaven (blue) linking with earth (red), and of course earth (red) linking back with heaven (blue). Taeguk basically symbolizes that the whole universe is in harmony. The taeguk circle stands for the eternal principle that everything in the universe is created and develops through the interaction between yin and yang; thus it symbolizes creation and development.

For more information please check out these links on the following topics:
Bagua
Taegeuk
Flag of South Korea
I Ching (book)