The center of the body - the Dantian
The energy center of the body has some properties that make it important for handling energy and feelings.
This center is recognized in various traditions. In Chinese Qigong it is known as the Dantian, the Japanese name is Hara, and in the Sufi tradition and the Diamond Approach it is known as the Kath. Since the name Dantian seems to be most commonly used, I'll refer to this center with this name.
The location of the Dantian is in the lower belly, 11½ thumb widths below the navel, 2 to 3 thumb widths inward.
Sensing the entire body
When you focus with your attention on the Dantian, you can simultaneously sense your entire body, from top to toe. It is this property that makes this the center of the body. Because of this, it is also easy to direct energy to a particular part of your body, when you have your attention at the Dantian. This is one of the reasons why in martial arts (like Taichi) the practitioner keeps his attention there. It also makes it possible to feel all kinds of feelings that are related to your body.
Sensations in the belly
There are particular sensations that can be felt at the point where the Dantian is located. It is where you can sometimes feel butterflies in your belly, like when you're in love. Somtimes you can feel a sensation of heat there, like when you need extra energy, or feel stressed.
As there are so many things about your body that you can feel in or from the Dantian, it is so that to be in touch with your body, you need to be in touch with your Dantian.
The Dantian can be used to meditate on. To practice this, you basically sit in an erect posture, and keep your attention on the Dantian. When you notice that your attention has drifted away, which is natural to happen lots of times, you gently return your focus. Do this for 15 to 30 minutes. You might want to check out the article on meditation.
There are also meditations that go a bit further than just focussing the attention. These attempt to store energy in the Dantian, or do even more esoteric things. These are considered Qigong exercises.
Moving from your center
In many Chinese and Japanese martial arts, the practitioner not only keeps his attention at the Dantian to more easily move his energy, but also to be able to move from the center. By letting one's movements come from the center, they are more easy, and have more strength and power.