Tai chi is a Chinese system of exercises that grew out of a martial arts style that was popularized by two families, Yang and Chen. Today, the widespread popularity of tai chi has lead to a number of ways to learn the movements for free.
The best way to study tai chi for free is also the traditional way: finding a teacher and studying under his tutelage. This is not as hard as you might think. Most major metropolitan areas have a concentration of Chinese people, and some Chinese communities, like those in New York and San Francisco, are known internationally. If you go to public parks in these communities, you will find traditional practitioners of tai chi. The best time to find them is first thing in the morning, from dawn to mid morning. They may have even formed a small club. Remember that in traditional culture, money is not usually the form of payment. You will pay your teachers with dedication, diligence and deference.
Literature and Videos
Tai chi forms have been recorded for many years. Many books and documents have been translated into English and are readily available through most public libraries. Videos will allow you to see the movements unfold, a distinct advantage over a series of still photographs.
There are a number of online resources that provide videos as well as documentation. You can access most of these sites for free. One advantage of using online sites is the forums, where you can discuss tai chi with other practitioners and get answers for any questions you may have. You may even be able to locate other practitioners in your area.