Partner training is more commonly known as 'pushing hands' or 'push hands'. These terms are actually quite misleading, because the exercises should have nothing to do with pushing, and nothing to do with the hands. This misconception is frequently interpreted literally, with practitioners just trying to push each over with the hands and arms, and that is not Taiji.
Well, to begin to understand this clearly, one must first completely discard the concept of trying to 'push' the other person, or to control them in any way whatsoever, and also to avoid being overly focussed on the contact points, such as the hands. Applying this understanding beyond mere 'lip-service' is much more difficult than it sounds because our current habitual reactions will not be capable, and new habits need to be cultivated.
In fact the old definition was "an exercise to sense and feel". The purpose of the exercises is to cultivate the awareness and sensitivity to changes of movement, position and direction; to develop an understanding of what is happening, moment by moment; and then to adjust one's own movement and timing accordingly.
The key skills to develop in partner training are:
These principles of how to train are explained in the Tàijí Classics.
All aspects of Partner training should be regarded as a learning opportunity. In order to learn, both must setup and cultivate the environment for each other to train. You must do your best to discard your old habits and to use only the Taiji principles without resorting to brute force, shoving and wrestling, leaning or 'digging in', regardless of whether you yourself are unbalanced or thrown.
There is more that could be said about this very important area of practice, but a long explanation will do little to enlighten the casual reader and will likely result in confusion. The best course of action is to join a class somewhere, and learn from a good teacher.
To properly understand Tàijíquán it is essential to study principles in the Tàijí Classics, and apply their interpretation into your daily practice of exercises, Forms and partner training.