Although there are different ways to practise tai chi, the importance in training one's tendon is universal to all serious tai chi practitioners. Tendon conditioning usually means lengthening and strengthening one's tendons, in particular tendons around one's shoulder joints and hip joints. There are however lots of confusions around the practice of tendon conditioning in tai chi.
The main issue is: How is tai chi tendon conditioning differs from normal stretching?
Before I tackle this question. Let's see how important is tendon conditioning in tai chi. In tai chi classics muscle training is usually associated with external martial arts, while tendon training belongs to tai chi, in particular as it is the power behind Fa Jing (發勁), or the externalization of internal power. So much so, some tai chi purists will insist that only tendon conditioning belongs to tai chi. They only train their tendons and refrain from training their muscles. They practise tai chi pushing hands with only tendon fa jing (also know as long jing 發長勁) in which people will be pushed to "fly" but without harm. It works on condition that both are of high skill level and both are focusing on tendon connectedness during practice: i.e. prepared to act and prepared to receive the act.
My training philosophy is that every training program shall be structured according to one's training objectives. And for most people (in both training for health and combat) both muscular training and tendon training will be needed. Muscular training in tai chi is similar to core-muscle training and the power base is our Dantian. Tendon training on the hand is training the connectedness and power of our tendons. The combat execution of the former includes short jing or more popularly called inch-punch (as popularized by Bruce Lee) which is useful in close combat.
Now let me get back to the issue of stretching versus tendon training. Both involve lengthening our tendons. In tai chi tendon training there are more to it. Firstly, the lengthening mechanism is spherical in nature rather than two dimensional and secondly, tendons in our body should have interconnectivity (the internal sensation is like chi running from tendon to tendon) and thirdly the muscles around the tendons will vicariously be trained too (which involve aligning or fitting our joints). However, the above features also create a constraint: our joints will not be conditioned to as flexible as in conventional stretching training.
(The above is a set of condensed notes to the subject)