Tai chi is the best known Chinese internal martial art, the other main ones being Bagua Chang and Hsing Yi. The emphasis in tai chi is not on muscular strength but instead on using the mind to leverage a relaxed body. There is a good essay on the differences between internal and external martial arts here.
Tai chi is suitable for people of any age and fitness level and unlike many other martial arts there is little risk of injury. The slow, flowing movements are a low impact way of improving flexibility, muscle tone, balance and coordination, studies having revealed numerous health benefits. At a high level tai chi can also be used for self defence.
The exact origins of tai chi are shrouded in mystery. According to legend it was invented by a Taoist priest called Chang San Feng who was inspired by a fight between a crane and a snake. Others credit the origin to Chen Wan Ting (1580–1660), a Ming dynasty general from Chen village who synthesized his knowledge of fighting strategy, Chinese medicine and Taoist yoga to create the art. Still others will point to Lee style tai chi as possibly being much older, perhaps dating from 1000 BC.
Whatever the truth of the matter, all the tai chi styles contain similar movements which are highly beneficial. For older people the high stance Yang, Wu and Sun styles are especially suitable whilst more younger and more athletic people might find more of what they’re looking for in Chen style.