Essentially, "Reiki" (pronounced "Ray-Key") is a term originally used to refer to a system of healing and self-development created by Mikao Usui in Japan in the early part of the 20th Century.
Simply stated, Reiki can be said to be a synergistic combination of 'spiritual healing' and 'energy healing' techniques.
Technically, the term "Reiki" is properly solely used to indicate the therapeutic and self-development system created by Mikao Usui (or, as he is often respectfully referred to as: 'Usui-Sensei').
More specifically, the name "Reiki" identifies the wonderful therapeutic energy radiance, or phenomenon, which lies at the heart of this natural healing system.
The "Rei" part of the name is often translated as "Universal*", and in essence, it refers to something Spiritual or Sacred, and can mean "Soul".
"Ki", is often translated simply as "Life-Force Energy", yet also implies 'Spiritedness' or 'Feelings'.
Thus, depending on ones perspective, Reiki can be understood as: "Universal* Life-force Energy", "Spiritually-influenced life-force Energy", or even as "Charismatic Healing Radiance".