I think the previous answers are good but there's more to it..
The dharmakaya, sambhogakaya, and nirmanakaya, together called the trikaya are the key summary of Buddhism and in fact Taoism and the whole spiritual shebang.
It encompasses all virtues, all powers, and all the things that you could possibly want.
One can work on each body and still not gain the others.
They are interpenetrating but they are also worked on individually.
This means that you can be Enlightened but not have any powers (effects of sambhogakaya and nirmanakaya).
I like to describe the Trikaya through the powers that one gains from perfecting them.
Body of Essence: Dharmakaya: being free from birth-and-death, no ego, disillusionment
Body of Virtue: Sambhogakaya: all the paramitas completed, all virtues completed
Body of Manifestation: Nirmanakaya: self-duplication (think Naruto), thought emanations, thought manifestation
The Zen masters all had the dharmakaya but usually did not complete the sambhogakaya or nirmanakaya. They were just like "Ah, I got everything. Worldly people and their affairs don't matter to me. Everything is emptiness. I am not real and neither are they. All that matters is my true self, which is indescribable. Off I go." But there was more to the Buddhist project than that!
Some Mahayanist masters on the other hand had the dharmakaya and the sambhogakaya, they exemplified stellar character yet also had deep realizations of the soul that the Zen masters had. They had achieved dharmakaya and sambhogakaya.
Lastly, some Taoist masters have the nirmanakaya but not the previous. They could perform miracles, thought manifestation, etc. Many could even live for thousands of years at a time. Nonetheless, they held too strongly to their notion of a self and did not see their true nature, their true self which exists beyond the body and mind.
But they could live forever, so that's something isn't it?
The Buddha said that these people are living the wrong path and he could see deep into their future lives and said they would suffer repercussions in the future lifetimes for it. (Of course you and I don't any of these three so we'll suffer even worse repercussions I think! So don't be looking down on these immortals with this piece of knowledge you have just attained.)
Many seers and people that can do things with their chi energies?
They are all just playing with the nirmanakay body.
Ever mantra and feel like the Buddha is helping you? That's a nirmanakaya manifestation. A whole bunch of them.
A Buddha actually completed the cultivation of Trikaya and stays in a separate dimension and sends out his nirmanakaya manifestation to people based on causes-and-conditions.
Big secret #2:
We all actually have ALL three bodies, too. They are a range and you can see in your experience how much of each body you have achieved.
Each can be summarized respectively in the following qualities: true awareness, true virtue, true power.
As you climb the spiritual ladder, you will develop these qualities but with the Trikaya model and the other models in Buddhism, you can turn it into a much more fun project and work on what you're missing... in a very long-term sense since usually nobody accomplishes all three Trikaya in one lifetime.
For more on this subject of Buddhism with this kind of thinking read "How to Measure and Deepen Your Spiritual Realization" by Bill Bodri.
Definitely, know that the Trikaya are NOT just some psychological thing. They involve things like spiritual immortality, immortality of character, and immortality of the flesh-body (respectively).
They umbrella the entire scope of development of everything with of course importance placed on the development of Buddhist emptiness for spiritual immortality and virtue and Taoism for cultivation of the power of multiple body emanation (yang shen).