Tai chi (or Taiji) is a physical art from China, which haw been practiced there for hundreds of years, and in the West for many decades. While it is technically a martial art, it is also a set of range of motion exercises. The focus in doing these exercises is meditative in nature, with a strong focus on proper body mechanics, proper posture, letting go of unnecessary tension, and moving as efficiently as possible.
The slow nature of the movements combined with the strong focus on proper alignment and posture make taiji very helpful for developing balance or regaining a measure of balance after injuries. The movements are widely varied, so that it is easy to develop within whatever current limitations one may be experiencing and onward to whatever level of balance and flexibility one wishes to reach.
Even though one does not require any significant degree of flexibility to do taiji, the regular practice of the motions has the effect of gradually increasing the range of motion in which one can move, and as the strength of the muscles increases, the flexibility does as well (recent studies have shown that although stretching does not actually increase the length of muscle fibers, the consequent strengthening of the opposing muscles DOES increase flexibility and range of motion).
Pain and inflammatory disorders
In the case of autoimmune disorders, pain disorders (such as fibromyalgia), or inflammatory disorders (such as arthritis), clients are often told to "exercise, but don't overdo it." This is a hard balance to hit, but due to the gentle and curving nature of the movements, taiji is a good candidate for such exercise.
Because taiji does not require equipment to set up or take down (you only need two feet and a floor, and as much space "as a cow could lie down in," it is very easy to do taiji in small increments. With no time invested in traveling to the gym, changing clothes, or getting equipment out and put back away, it becomes very easy to do taiji in short increments. If one has pain or fatigue issues, and can only manage 2 minutes of activity, it is quite a simple matter to get up, do two minutes of taiji, and sit down to rest. In this way, one can gently accumulate a whole lot of gently and helpful exercise through the course of a day. Obviously, if one has medical issues one should CHECK WITH A PHYSICIAN to be sure that there is nothing in your own physical issues to prevent you from safely doing any physical activity.
The quiet and focused nature of taiji makes it a very useful form of meditation as well. The slow movement with focus on particular aspects of what one is doing is a great example of the "here and now" focus of a meditation exercise. The regular practice of this combination of gentle activity with the practiced focus helps to develop attention and focus which will be useful in all other areas of life, as well as a stress management tool and a tool for managing anxiety disorders.
Stye or School
There are many styles of taiji (the most common being Yang, Chen, Zhaobao, He and Sun), and the devotees of any particular style can be found to tell you why their own style is the best / only taiji worth doing. The reality is that all are fine, and the best one is really:
a) the one you are most comfortable with
b) the one you can find teachers for near you.
Other than that, it really doesn't matter which one you do.
It IS best to find a class or a teacher though. you can easily find DVDs or YouTube videos to try to learn from, but a video cannot look at what you are doing, and give you feedback. A teacher can, and this will be enormously helpful in learning taiji. Even if you only get to class once a week, the feedback will make your practice much more effective throughout the course of the week between classes.