T’ai Chi Chuan is a complete system of exercise and a martial art based on the theory of Yin and Yang. Don’t worry about it being a martial art, you won’t be expected to fight, at least not unless you really want to!
There are many myths and legends about the origins of Tai Chi. Most claim it came from Wudang Mountain where, in the about 12th century , a Daoist monk, Chang San-feng, was concerned that the novices were lazy and unfit compared to those at the Buddhist Shaolin Temple. After watching a fight between a snake and a crane he had a dream which lead to his creating the art.
There are numerous equally unlikely variations and other completely different stories such as the Chen family claim that it was invented by one of their ancestors. In fact it almost certainly developed over many generations but may well have a connection to Wudangshan and the legendary monk as well as some input from the Chen’s.
The name taijiquan is not even found in the historical record until the middle of the 19th century when Yang Luchan arrived in Beijing and popularised it, first as a martial art, and then as a health giving exercise system for the elite.
My own teacher, Dan Docherty, has done much research into the origins of Tai Chi and has come to the conclusion that we can never ‘find THE truth’ and that, as an art, it has and will continue to change.
What matters is whether a style accords with the Tai Chi Classics, is effective as a martial art and has a lineage back to Yang Luchan. If not then it may be an internal art but it is not Taijiquan.
There are also many, many other internal martial arts in China – Xingyiquan and Baguazhang to name just two that have become popular in the west.