Tai Chi, as it's practiced in the west nowadays, may perhaps best be thought of as a moving class of yoga and meditation blended. There are a number of supposed forms (occasionally likewise called 'sets') which consist of a succession of movements.
A lot of these movements are originally deduced from the martial arts (and maybe even more ancestrally than that, from the innate movements of creatures and birds) while the way they're performed in Tai Chi is slowly, gently and gracefully with fluid and even transitions between them.
For a lot of practitioners the focus in doing them isn't, most importantly, martial, but as a meditative exercising for the body. In Chinese philosophy and medicine there lives the concept of 'chi', a life force that animates the body.
Among the professed aims of Tai Chi is to further the circulation of this 'chi' inside the body, the notion being that by doing so the health and life force of the individual are enhanced.
This 'chi' mobilizes in patterns that are closely related to the nervous and vascular system and therefore the notion is closely affiliated with that of the practice of acupuncture and additional oriental healing arts.