Taijiquan, and the Chinese martial arts in general, do not go in for a lot of saluting, bowing or other rituals, compared, say, to Japanese traditions. This is not to say that respect is not important in the training hall, it is just that the Chinese way of doing things is a bit more low-key.
It is important, however, as with everything we do during a practice session or class, to behave mindfully, and therefore it is worthwhile to learn to salute well; giving a salute is one of the first things that we do and doing it properly will help to "nudge" you into the right frame of mind. Moreover, if you ever meet teachers or students from another school or who are senior to you in your own lineage, your salute will say a lot about your attitude and awareness to those who are attuned to body-language and posture - so it is a good idea to make a good impression!
How to salute:
Circle the arms up from the sides to the level of the middle of the chest. Make a fist with the right hand and place it with its face into the palm of the left hand. The fingers of the left hand are extended so as to be flat and the thumb is held at the side of the palm. It is important that the thumb is not held stuck out, as this signifies that you think that you are "number one" and is not a very polite way of greeting somebody!
The right fist represents the Yang aspect, while the left palm represents Yin; thus the salute is a reminder of one of the fundamental principles of taijiquan, though it is common in other martial arts as well. Some also suggest that the aggression of the right fist is being restrained by the gentler palm and therefore the salute is representative of a peaceful intent.
While making the salute, the arms should be held in a circle in front of the upper chest, with the shoulders relaxed, but care should be taken not to let the elbows droop or the salute will look lop-sided and/or sloppy. It is also important not to reverse the hand position inadvertently as a left-fist/right palm position is associated with mourning and would be considered very inauspicious in any other context.
You will see subtle variations on this posture and there are also many other symbolic ways in which it can be interpreted (see, for example, here), but this simple version will suffice to begin with. When learning tai chi, it is important to pay attention to increasingly subtle details, so learning to salute correctly is a good way to start.
Andrew Howard is a Chen-style Tai Chi Instructor teaching in West Dorset, UK.