The main idea is that energy is flowing through your body, and is what's keeping us alive.
The therapy consists on using hands to guide that force of energy inside us. They say if that energy is low, we are propense to feel bad and when it's high, we are healthier and happier.
Reiki is a japanese combination of "Rei" which stands for "Greater power or even god's wisdom" and "Ki" which means "Life energy" so the idea is that reiki is a "Spiritually guided life force energy".
Invented in 1922 for Mikao Usui, a japanese budhist, has been used universally and their practicants believe they can promote emotional or physical healing. But this "energy" has never been demonstrated by scientific methods.
While Reiki is spiritual in nature, it is not a religion. It has no dogma, and there is nothing you must believe in order to learn and use Reiki. In fact, Reiki is not dependent on belief at all and will work whether you believe in it or not. Because Reiki comes from God, many people find that using Reiki puts them more in touch with the experience of their religion rather than having only an intellectual concept of it.