Open and Close (開合) in an important practice concept in Tai chi classics. Like other classical concepts, it is subject to different interpretations, by different practitioners. I offer mine here. And I shall discuss this concept as it is applied in doing tai-chi form, of whatever style (though my own practice is Wu-style).
For every sport and physical activity using the upper body and executed through one's arms, the power-house is always one's shoulder blades (夹脊). Force is being transmitted from the shoulder blades through a firm foundation (created by sound foot-stance), and executed via a secondary power-house: one's pelvic joints, focused and balanced at the tip of one's vertebrate (尾閭). The latter acts as a pivoting point for the body's turn while the former acts as a pivoting point for the arm's swing. If you observe a discus thrower or golfer in slow motion, you can perhaps better understand my explanation here.
In tai-chi, Open (開) simply means opening these hinges (outwards, away from the center) whereas Close (合) simply means closing these hinges (inwards, towards the center). The question is: How is the practice of tai-chi form differs from the training of other sports or physical activities using these hinges?
Tai-chi form is to be done in slow motion. When a practitioner does his form, he should be in a semi-meditative mode, focused his mind on the relevant hinges during his movement. For example, when his moves his arm outwards (opening) as in the preparative part of "Brush Knee Twist Step" 摟膝拗步, he should focus primarily on opening the shoulder hinge and secondarily on opening the pelvic hinge. And when he then pushes his hand out in a closing mode, he should likewise focus on closing the hinges. And these are heavy hinges in his mind, serving the objective of a work-out (Jing-training 練勁 or Jing-generation).
What is the criteria of success? In doing the above, if one can feel continual resistance at the joints when the hinges are being opened or closed, one has successfully executed the Open/Close concept in doing his form. And it makes all the difference in one's understanding of tai-chi, and one's tai-chi work-out will be more intense as well as more fruitful.
One final advice: in doing tai-chi normal breathing is usually sufficient. However, if one wants to synchronize his breathing for a more fruitful workout, he can try doing breath-in when Opening and doing breath-out when Closing.