Being centered means that one is comfortable being present, one is relaxed, and one is balanced. One is in a state of rest mentally, emotionally, and physically when one isn't doing something.
A person who is busy reacting to the room and the circumstances, who is thinking compulsively, who is physically uncomfortable and fidgety, won't be able to give nearly as much attention to helping somebody as would a person who is centered. The fidgety person might be just as knowledgeable and able, but if she wastes her energy distracting herself she won't get much else done.
In martial arts one of the first things one will be taught is to be centered while doing nothing. Old Chinese Tai-Chi masters would for the first many months only teach their students how to stand. Only when the students could stand correctly, that is, in a centered, balanced, grounded way, then they would go on to learning moves.
We could very well apply the same principle to processing. We could spend the first months teaching you how to sit still in front of another person without having to do anything. And only when you could remain perfectly balanced would we go on to teaching you what to do with the person in front of you. That would be a perfectly valid way of doing it, and it illustrates the importance of being centered.
However, westerners tend to be more in a hurry, so we aren't going to wait for you to become perfectly centered before we go on. You should start doing centering exercises right away, and you should continue doing centering exercises as you go through your training and as you work with clients. In time you will become a master at being present in a centered way.
Centering exercises are a type of meditation. However, they are not particularly done to advance the person's consciousness, even though they do. The intention behind them is simply to get a person to be comfortably present as preparation for taking action.
It may sound as a paradox, but one is more able to do something if one can do nothing first. One will walk better if one can stand, one will speak better if one can be silent, one can think better if one can start from a quiet mind. An action or a communication is more pure and clear if it comes from a point of balance.
The basic centering exercise is simply to sit down comfortably with closed eyes and do nothing but being aware of what is going on. Don't try to do anything in particular, and don't try to NOT do anything in particular. Just notice what is happening, what sounds are in the room, how your body feels, what thoughts go through your head, and so forth. Don't try to change or stop any of it. Just perceive it all as naturally occurring noise. Gradually your mind and your body will relax more and simply allow everything to happen. And your thoughts and your feelings will go quiet.
If your mind is too busy it can be a help to relax it if you pick something specific to focus on. In meditation one uses mantras, which are non-sense words or sounds that one is repeating over and over in one's mind. That keeps the conscious mind busy and it forgets to grind away at other thoughts. That is a perfectly valid way of getting one's mind centered. However, eventually one needs to be able to sit down and be centered while having one's attention outward and without doing any special tricks like that.
Once one can stay centered while sitting still by oneself with closed eyes it is time to raise the stakes. Since all of this is preparation for working with people we will need to introduce some people. A centering exercise one can do with another student is to sit down in front of each other, just looking at each other without doing or saying anything. As with any other centering exercises, one will find at first that various phenomena will happen. One will react in various nervous ways, one's eyes will water, one will doze off, or whatever. As one continues to do the exercise the phenomena will tend to taper off and eventually one is able to do it while staying centered all the time.
Once one can stay centered in front of another person doing nothing, one can then work on staying centered in front of somebody who is doing something. Particularly somebody doing something that invites or provokes reaction. If we train a student to remain balanced in all kinds of ridiculous situations chances are that she will remain balanced while working with normal situations. So, we can instruct another student to improvise statements or actions intended to break the student's balance. And every time the balance breaks and she laughs or twitches or something, we will continue repeating the same behavior that triggered the reaction until she succeeds in remaining centered under all conditions.