According to Bruce Frantzis (a well known Tai Chi Master) in an appendix section of his book the Power of Internal Martial Arts on the history of Tai Chi, the answer is as follows (it's quite long so this is a paraphrased version, for the longer one you need to read the book):
Tai Chi's history can be broken into 2 parts
Pre and Post Chen Village. The Pre Chen village is covered in obscurity and speculation, the post Chen Village timeline is quite clear.
Since all current high level Tai Chi in China came from the Chen Village, I'll discuss the simple one first.
The Chen Village possessed a phenomenally powerful martial art style which allowed them to prosper. For reasons (presumably to maintain their village's status), this art was kept secret and outsiders were forbidden from learning or entering the village. A young man named Yang Lu Chan was an avid martial artist student, who eventually reached the point where his teacher felt he could teach him no more, and recommended him going to the Chen Village to learn Tai Chi.
Being extremely motivated, Yang pretended to be deaf and was accepted as a servant eventually becoming trusted enough to gain the keys to the inner training area. In the evening when current master Chen Hsing was training his students, Yang would observe the class from afar listening intently then once it ended practice the material late into the night before waking up early in the morning to begin his normal job as a "deaf" servant.
After 6 years, Yang was caught, with many of the Chen village members wishing to throw him out or kill him on the spot. Yang managed to be given the opportunity to demonstrate his abilities, and was able to easily defeat each other student, arising Chen Hsing's curiosity. This in turn led to him testing Yang for suitability as a future disciple (as Yang has already demonstrated extraordinary perseverance and discipline), and once he passed Yang spent the next 18 years training intensely, at which point Chen Hsing gave Yang Lu Chan his blessings to spread it to the outside world.
Once left the Chen Village, his martial skills were at a phenomenally high level and he was able to defeat every opponent, or groups of attackers that came at him with ease, leading to him becoming known well enough to be recognized by the Emperor as the top martial artist in China and get the job that came along with this distinction. Yang also taught regular citizens, and his top disciples passed on the art allowing it to spread throughout China through the ages, primarily with Yang's style and one originating from a top disciple, the Wu Style one designed with an emphasis on healing (a third style called the Hao style also exists).
After Tai Chi was introduced to the Chen village, they spent over a 100 years developing and perfecting the art but how it first arrived there is a matter of speculation. 4 main theories exist to explain the pre chen period, each of which is often cited as the authoritative source for Tai Chi's history but no reliable evidence proves one over the others.
Theory 1
A famous taoist immortal (one who had achieved true enlightenment) named Chang San Feng is often credited with having developed Tai Chi, although this has not been proven. The classical story goes as follows:
Tai Chi as we know it was first developed in the Ming dynasty, by a young Shaolin ex-disciple named Chang San Feng. One day, he was outside his hermit hut, while he noticed an eagle grappling a snake. He noted that:
"If the eagle strikes the head of the snake, the tail whips around and hits the eagle. If the eagle attacks the tail, the head twists around and bites the eagle. If the eagle strikes in the middle, both ends strike."
He noticed that, albeit the small size of the snake, in comparison of the eagle, it twisted and turned in the eagle's claws, thus not allowing the eagle to strike. The fight was soon over, the snake victorious.
Chang San Feng then went on to develop Tai Chi based on the smooth, liquid movements of water. He "found the harder you pushed a ball into water, the higher it bounced back.
Chang Seng Feng's new martial arts style differed from those of the era, and eventually was passed down to Wang Tsung Yueh who randomly arrived one day in the Chen Village (an odd occurrence since it was in the middle of no where and far away from the Shaolin Temple).
[pause theory 1]
Theory 2
Wang Tsung Yeah learned tai chi from one of many existing Tai Chi lineages (possibly Chang San Feng or even an older one) and randomly arrived at the local inn [resume theory 1] full of closed Chen Family insiders who held a great deal of pride in their village's martial arts. Wang disparaged and then insulted the village's martial arts, which provoked them into attacking Wang only to be completely defeated. In awe of his abilities, they begged him to teach them-but lacking the time to fully teach the art he instead adapted its energetic components into their Shao Lin Cannon Fist style.
Theory 3
Tai Chi was created by a resident of the Chen village, Chen Wan Ting who was a top general in the Chi Chi Guang army. Chen created Tai Chi through combining elements of Chinese Medicine (especially the meridian line theory) with Chi Chi Guang's Shaolin martial art methods. This theory is largely supported by the overlaps between Chi Chi Guang's military manual and the basic techniques in the Chen Style.
Theory 4
Chen Wang Ting (still a general but belonging to a different army) gave Jiang Fa political sanctuary in the Chen village in return for him teaching Chen's clan Tai Chi Chuan. Jiang Fe happened to know both Wang Tsuang Yeah's energetic power system and Chi Chi Guang's martial arts methods (which were basically a compendium of the best existing Shao Lin techniques). It is not known how Jiang Fe came to possess this knowledge, but after he imparted it to Chen Wa Ting, creating the First Set of Chen Style Tai Chi, Chen Wa Ting then later developed the second half (Cannon Fist) on his own.
Although Bruce believes the history is too vague to ever be certain of the exact origin of Chen Tai Chi two facts are clear:
a) Chen Tai Chi partly derived from Chi Chi Guang, there are too many similarities between the two.
b) Chen Tai Chi's energetic components were too advanced to have just been created through an adaption of meridian line theory (they are too simple).
Hence you can infer each theory is partly but not completely true, which would also explain why each still has many adherents.
http://wiki.answers.com/Is_there_a_website_that_teaches_tai_chi_for_free
According to Chinese tradition, Tai Chi Chuan was invented by Shang Sangfeng in the Wudang Mountain monastery, sometime in the 13th century. However, official imperial records, dating to the Ming Dynasty, state that Tai Chi Chuan was invented by the patriarch of the Chen family sometime in the early 17th century. Tradition says Sangfeng, official Chinese history says Chen family, hard to say which one is true. However if I had to venture an opinion on the subject, its not very humble of the Chens to run around claiming credit. Supposedly Tai Chi is Daoist, and Daoism demands obscurity and humility, not running around announcing things and taking credit for things. In Daoist humility fame is earned by merit, not running around screaming in the streets "here's a document proving my family invented it!"
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