Square form: in doing thee square form correctly, a student must carefully align his structure, in particular those joints responsible for the next movement (primarily shoulder and/or hip joints). Like shooting an arrow a slight movement to the opposite direction will be beneficial to initiate the movement. The internal sensation is finding the points of maximum resistance while the objective is to open one's joints (and strengthen it at the same time). Analogy in Chinese calligraphy is 楷書 (regular script)。
Round form: This presupposes a prior training of square form. With prior proper joint opening/strengthening, round form aims at "rounding the angles in the joints" (as per points of joints being aligned during square form). The movements themselves will look simplified when compared to the square form. Without prior training in square form, round form will look structurally weak. With prior training in square form, the curved angle will have power to deliver a strong rounded structure. Analogy in Chinese calligraphy is 草書 (cursive script)。
Fast form: The fast form as it name suggested is to be executed (visually to an outsider) faster than both square and round forms. Prior training in square and round form is required. Its form is similar to round form but can be much more cursory and is practised in such a way that a sudden explosion of power while doing like the round form. The explosion of power is fast and quick. And the power comes from overcoming the internal resistance of our body (primarily our joints), and thereby trained our body's ability to externalize power with full-body connectedness. A practitioner here is free to choose which individual movements he wants to externalize his power for training purpose.
The above is the gist of tai-chi's training method as in internal martial arts. It is an excellent way to improve our health and strength for people of any and all ages. Needless to say, full-contact combat training (conditioning side) has to include elements of external martial arts (for example, should include hitting a heavy bag for strengthening one's "points of contact" and building up the line of power transmission between the body and the "points of contact"). But the fact is that those who are interested in full-contact combat will most likely not be interested in tai chi, and those who are interested in tai chi will most likely not be interested in full-contact combat. As a result, the training of modern tai chi will be sufficient using the three different forms (square, round and fast) as far as the interest of students is concerned.