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Tai Chi is a form of kung-fu, a fighting art developed in China over the last thousand years. You may be interested in History for more information.

The short answer is yes, the long answer is, not unless your teacher shows you how. You can spend ten or twenty years refining your form, but if your teacher can't even feel his chi, you never will. You need to find a teacher who can show you how to feel your chi within 100 to 200 days (3 to 6 months). Longer than this, and you are either not practicing every day or you are wasting your time with that teacher. After that, use the 3 to 6 month principle to judge yourself and your teacher. If you have not made progress every 3 to 6 month period, ask your teacher for advice. If a year has gone by of your own hard practice and you have not made any improvement in your skill, be honest with yourself, none? Nothing significant? Then talk to your sifu about it, if he has a good explanation that's fine, but otherwise you should find another teacher. As long as you are working hard, practicing your form 5 times properly a day or more, it is your teachers responsibility to guide your progress, and there is no reason you should not make continuous, regular progress. Anyone who tells you differently is trying to sell you something.

And if the school does not teach push hands (at all), or is insular (does not respect other lineages of taijiquan at all) do not waste your time with them. It's a safe bet a school is not insular if they teach two or more styles of Tai Chi, if they do push hands, if they ever compete at anything, or if they ever invite masters who are not directly connected to the school to give training seminars. Ask them if they ever take “trips to china” etc. is another clue. Generally you want a school which is not afraid to show it's stuff and is generally aware that there are hundreds of good tai chi schools in the world and that no, they are not the only ones with the good stuff. Humble teachers are famous teachers.

Qi (various spellings) is the internal energy of the human body. However, you have to be careful because saying 'internal energy' has just traded one term for another; now you need to know what internal energy is. You are probably looking for the Chi FAQ.

This means, what is the role of supplementary exercises like standing meditation and silk reeling exercises. Chen Zheng-lei explains,

In the old days, children would begin by learning forms right away, in the traditional manner of learning sequence, corrections to movement, and then developing internal energy. Now, things are more sophisticated; children begin training with flexibility exercises and “chan si gong” [exercises for training silk-reeling energy] but do not begin the forms for a while.http://taiji-bg.com/articles/taijiquan/t27.htm

Traditionally, everyone begins with Lao jia, Yi lu [old frame, first routine] at first. From there, they learn the single-edge sword [broadsword]. After training in Yi lu for two years, the body is relaxed and the student begins learning pushing hands and Er lu [the second barehanded routine of Chen Style].“