The Korean national flag is called Taegukki (from Tae Guk, or Tae Geuk, meaning the universe, Ki meaning flag). The symbol in the center of the flag is called Tae Geuk, and it represents the duality of the cosmos.
The blue side represents Um (or Yin in Chinese), which is negative energy, darkness, cold, earthy and female. The red side represents Yang (also Yang in Chinese), which is postive energy, light, hot, heaven and male.
Tae Geuk itself, with its balance of Um and Yang, represents harmony, and the harmony of Um and Yang is the Do (Tao in Chinese), the path or the way.
The symbols in the four corners are called Gwe, representing the principle of movement and harmony, and each is a trigram from the I-Ching, the Chinese Book of Changes. The three unbroken bars at the top left stand for heaven/Yang (Keon). The three broken bars at the bottom right stand for earth/Um (Kon). The symbol at the top right - yang within um - stands for water (Kam), and the symbol at the bottom left - um within yang - stands for fire (Ri).
Notice how Keon (Heaven) is the opposite of Kon (Earth), and Ri (Fire) is the opposite of Kam (Water). This again represents the duality of the cosmos.
Neither is the white background accidental: the color traditionally represents 'peace', and the peaceful Korean people of old were known as the 'white-clad nation.'