Tàijíquán is sometimes written as Tai Chi Chuan, depending on the method of Chinese to English translation. Sometimes it is referred to as simply "Taiji" or 'Taichi'. It is well known as a system of exercise, with a reputation for aiding relaxation, but what is not so well known it that Taiji is fundamentally a martial art.
The practice of Tàijíquán is based on a collection of principles known as the Classic Texts, or just "the Classics". They form the written advice from highly capable practitioners and teachers, over several generations, on how to practice and apply the art. Guided by this advice, we train to develop good mental habits, good body habits, and ultimately to understand ourselves and become better human beings. Tàijíquán is a system of mental/physical self development, grounded in the practice of an effective martial art, and it is not mystical or religious.
It is not so widely appreciated that Tàijíquán can, if trained appropriately, be used as an effective martial art. The emphasis is on using subtle whole-body synchronized movement with finely tuned awareness and timing, to overcome raw (disconnected) speed and brute strength.
The disadvantage of a 'soft' style such as Tàijíquán compared with 'hard' and 'hard-soft' styles such as boxing, Karate or Kung Fu, is that it takes many more years of practice to achieve a basic martial skill. The advantage of Tàijíquán however, is that development can continue far beyond the limitations of age, strength and speed that accompany 'hard' styles, and it is not injurious to the body.