Wikipedia
In the context of taekwondo the term taegeuk refers to a set of poomsae or forms used to teach taekwondo. A form, or poomsae (also romanized as pumsae or poomse) is a defined pattern of defense-and-attack motions. Outside of the context of taekwondo, the word taegeuk refers to the Taoist principle of the "unity of opposites" (yin and yang). Taegeuk is also the name of the red and blue circular symbol used in the flag of South Korea.
Between 1967 and 1971 Kukkiwon-style taekwondo made use of an older set of forms called the palgwae forms developed by the Korea Taekwondo Association (KTA) with input from some of the original nine kwans of taekwondo. By 1970 additional kwans had joined the KTA so the newer set of taegeuk forms was developed to better represent inputs from all the participating kwans. By 1971 the palgwae forms were considered to be deprecated in favor of the newer taegeuk forms, though some school still teach palgwae forms as well. All WTF poomsae competitions use the taeguk poomsae, along with 8 of the black belt poomsae.
Each Taegeuk form symbolizes a specific state thought to be indicative of the belt the student currently holds, and is represented in WTF Taekwondo by trigrams (originally derived from the I-Ching) similar to those found in the four corners of the South Korean flag.
Taegeuk (, ) refers to the universe from which all things and values are derived. It is also the symbol that makes up the center of the flag of South Korea and the source for its name, taegeukgi (hangul: 태극기, where gi means "flag"). The taegeuk is commonly associated with Korean Taoism philosophical values as well as Korean shamanism.