First, the name Taijiquan (Mandarin pinyin spelling) is not all that old.
The common account of the origin of the art is NOT Zhang Sanfeng. That story has been bandied about for a while but was actually introduced for the first time about 100 years ago. Before that, no one spoke about such origins.
The most common KNOWLEDGABLE beliefs are that the art originated in Chenjiagou (Chen family village) approx. 400 years ago.
chen style is characterized by rhythm changes, spiraling and coiling enrgy, silk reeling, explosive power (fajing) generation that transitions back to soft power, etc…
It has gone through a number of iterations - old frame, new frame, etc…
Then about 150 years ago, the Chens taught a man named Yang Luchan (one of their servants - there are many stories about how he became a servant but the one that was listed BEFORE the rewrites of histoyr had him as an indentured servant - a common thing for poor families to ‘sell’ their children as servants t make sure they at least survived.
Yang was given his freedom and set up with a job at an herbal shop as an assistant when the elder patriarch of the chen family died. He had been a good servant but also, at that time it would have been very improper for the widow to have a male servant NOT a eunuch… so, since Yang had been agood servant - the family did right by him.
He taught several places, developed a reputation, and ended up in Beijing. He taught Wu Yusiang (spelling may be off)… Wu later went to the chen village and learned some more. combining the two methods (Yang’s already changing style from the Chen family and the other Chen methods from his second Chen teacher), he created the Wu (martial spelling as in Wushu) style. This style has two main branches now but is commonly referred to as Wu/Hao style since Wu taught Hao Weizhen who carried it forward.
Yang Luchan and family later taught Wu Jianquan (there are issues about was it Luchan or his sons but the common belief is that the sons would have been listed as his teachers out of formality due to royal students having to be a higher generation than Wu…
Wu Jianquan took the Beijing frame he learned and devise the Wu style. The spelling or character for this Wu is NOt the same as the earlier wu.
Hao Weizhen was befriended by Sun Lutang. Sun was already reputedly a high level master of Bagua and Xing yi. After helping Hao recover from an illness, Hao taught Sun his Taijiquan. Sun then combined it with his Bagua and Xing yi and created Sun style Taijiquan.
There are other branches as well…but these are the main ones.
They all have certain main principles in common…such as distinguisning between full and empty, one part moving up when another is moving down, one back - the other front, dropping the shoulders and elbows, moving from the center of mass, the use of the torso and waist,…etc…
so, many say that as long as you adhere to the common principles, there is only one Taijiquan…a bit simplistic because they all have unique flavors and strategies. You simply can’t do Sun style with the same attitude as Yang or chen like Wu…
There are also newer routines…some of the new ones are quite good…and others are quite bad.