Tai Chi is the urbanization of the phrase T'ai Chi Ch'uan (Taijiquan), meaning "Supreme Ultimate Fist (School)" and usually refers to the internal martial art of that name originating in China.
An internal martial art is one that focuses upon the energy, chi(qi), of the body and its application for martial purposes as compared with an external martial art such as karate, which emphasizes muscular strength.
Taiji is commonly characterized by slow, flowing movements interspersed with rapid ones, and emphasizes posture, rooting, balance and efficient body mechanics for both health and martial applications.
The history and lineage of Taijiquan is often debated, but most discussions place its origin in the late 14th - early 15th century from one Chang San-feng. Its philosophy and movements have their roots in Taoist philosophy, itself dating back at least to the 6th century BC. Chief among the main concepts are relaxation in movement, tranquility, proper body alignment, balance, and focus upon both breathing and chi(qi), the internal energy utilized in Taijiquan.
A sister practice, chi kung (qigong), or "skill in working with energy", utilizes the same principles and movement techniques as Taijiquan. An emphasis upon health benefits, breathing, energy flow and posture characterize qigong. Although there are some qigong styles that include martial applications, qigong is usually practiced in the West as a health exercise.
As for the practice of Taiji in the West, it too is often focused solely upon health benefits, but an in-depth education in this art involves both martial and health applications. The benefits to be gained in this education start to appear quickly, but the study of the art can take a lifetime.
The T'ai Chi Master makes those complex movements appear easy.
See tai chi, qigong