Reiki is an increasingly popular form of energy healing, in which a practitioner channels healing energy through his/her hands into his own body, or the body of a client. Although not officially a Buddhist practice or form of healing, historically Reiki does have Buddhist roots, through its creator, Japanese Buddhist Mikao Usui.
Usui lived in the late 19th and early 20th centuries (1865-1926) and was a Tendai Buddhist. Tendai Buddhism has strong roots in Shintoism, the pre-Buddhist religion of Japan, which is heavily based in nature, and ascribes spirits and spiritual energy to all aspects of nature - elements, plants, and animals. Tendai Buddhism traces it roots back through China to Tibet, and shares many Vajrayana Buddhist sutras with some Tibetan traditions (although it is not usually itself categorized as a Vajrayana lineage.) Vajrayana Buddhism traditionally involves esoteric energy practices and symbol work, often including chakra work, and so there were likely energy practices involved in the Tendai Buddhism that Usui studied and practiced.
According to the traditional story of Reiki, Usui received the primary insights related to his Reiki system while attending a 21-day Buddhist retreat in 1922. This retreat most likely included meditation, fasting, chanting, and possibly energy work that was common to Tendai at the time. Some biographers claim that Usui actually was using some of the techniques now classified as Reiki before this retreat, and that they may have been rooted in Shugendo, a kind of Japanese Shamanism practiced in the mountainous area of Japan that Usui lived in, and that combined many Shinto and Buddhist ideas with hands-on healing methods. In any case, after his 21-day retreat, Usui began to formalize his work.
There is also evidence that later in life Usui studied Pure Land Buddhism, Shingon, and other Japanese Buddhist practices more directly related to Vajrayana Buddhism. All of these Buddhist lineages share the Mahayana Buddhist belief that enlightenment is possible at any time, and not necessarily the product of a long mulit-life spiritual process. This idea pervades Usui's writings on Reiki, and Reiki is viewed by some as first and foremost an esoteric practice for attaining enlightenment, with any physical healing benefits as secondary.
Another way that Usui drew upon Buddhism in the development of Reiki was by providing Reiki with 5 precepts, reminiscent of the 5 precepts in Buddhism of abstaining from violence, stealing, sexual misconduct, lying, and substance abuse. Although modern-day versions of Reiki's precepts vary, in general the five ethical principles on a basic level are to refrain from anger, refrain from worry, practice gratitude, work hard, and be kind.
Today there are many different branches of Reiki practiced around the world, most stemming from the two main branches of Traditional Japanese Reiki and Western Reiki.