Most of the Tai Chi you see in the West is a gentle exercise developed through the 20th century, based on a hardcore martial art of several schools and lineages, all deriving ultimately (depending on who you believe) from either ancient Daoist cultivation, or the martial art of a specific farming community, the Chen clan, in China.
What's commonly seen is a more or less watered-down offspring of the Long Form (continuous, connected exercise) of the earliest "forms" of Taiji, which were sort of like compendia of all the techniques on the art, strung together in a way designed to develop the right kind of co-ordination through repetitive exercise.
As a martial art, it was widely recognised in the China of the late 19th century and early 20th, as one of the most effective martial arts of China, a sort of national treasure - this is through matches of competitive skill, not in theory - in those days the spectacle of a famous martial artist setting up a podium in a public place and challenging all comers was not uncommon.
The principles of Taijiquan (Taiji fist, Taiji being the famous Yin/Yang diagram) that make it (and a few other arts with similar principles) stand out from the majority of other martial arts are twofold: in terms of body development (how power is generated) and in terms of actual overall fighting strategy.
In terms of body development, power is generated by a way of using the body that's rather unusual and has to be trained for a while independently before any really serious martial application of it can be considered. It's a complex new type of co-ordination that requires training out of the normal habitual way of moving one has picked up, to a way of moving in which the body's frame is continuously controlled as a unit, any movement of any attacking or defending part of the body being at the same time a movement of the whole body, utilising clever leverage principles which take advantage of the solidity of the ground, and transmit it through efficiently aligned bones, using the minimum necessary muscular energy. This hyper-efficiency makes Taijiquan eminently suitable for combat in war, which requires tremendous endurance.
In terms of strategy, Taijiquan makes use, again, of a "soft" or "yielding" type of strategy. The idea is to get in contact with the opponent in a way such that the opponent can't feel your intent, but you can feel theirs, through their body, and through their body's connection to the ground. The Taiji fighter then breaks their opponents connection to the ground, rendering them without the ability to gain any leverage, and therefore helpless, and easily killable by a type of focussed, explosive use of force called "fajin". (The aim of most Chinese martial arts was originally to incapacitate or kill the opponent as quickly as possible. A Bagua practitioner of the old school recently said in an interview: "In Bagua, one wants to make the opponent spit blood".) A poetic description of this combination of "softness" and "hardness" is "like a needle in cotton". Some poetic ways of what it feels like to fight a Taiji fighter I've seen are: like trying to push a beach ball under the water directly from above; like fighting a boa constrictor.
There is another interesting aspect to Taiji power than clever leverage though, and that is the use of the body's connective tissue, the fascia, as an additive element to the force generated through leverage from the ground. Training in these aspects of the art involving the fascia produce feelings in the body which have led credence to the idea of "qi" being an external force. What's actually happening is that control of the fascia (which has recently been discovered to have some muscle fibers embedded, and other hitherto-unrealised interesting properties which might be responsible for some of the documented analgesic effects of acupuncture) is acquired through a control of aspects of the autonomic nervous system, requiring a certain type of mental state akin to hypnosis. (Hence the requirement for some meditative training in many of these types of "internal" arts.) In the course of this training, one sometimes feels one's body to be held or moved in a kind of "magnetic field". It's easy to see how people could think this a real thing, but the correct explanation is probably more along the lines I've given above: it's an odd brain trick that makes the body in some sense feel "alien".
Needless to say, you won't get this kind of stuff from your local mall, but it's the sort of training that people can get if they look for it: there are perhaps dozens of teachers of the "real thing" scattered throughout the world, and concentrated in a few places in China, and maybe a few thousand serious practitioners.
