Reiki is a Japanese healing art. A Reiki “Master” is a translation from the Japanese word Reiki “Shihan”. Shihan means a respected expert teacher, “master instructor” (of either a martial art or a healing art, such as Reiki).
Just as in other Japanese healing traditions, it took a long time to become a teacher, and one had to practice the healing art as a Reiki practitioner/healer first. Once someone became a Reiki Shihan, that person could not only give Reiki treatments, but also teach Reiki. So to answer your question, a Reiki Master (in the traditional meaning of this word) is a Reiki healer, who is also a Reiki teacher.
In the West, the translation “Master” has shifted from the concept of Shihan “master instructor” as a “teacher” to that of “master” as an “expert”. Many Reiki practitioners today refer to themselves as Reiki “masters”, whether or not they teach Reiki, which is technically incorrect. Also some Reiki teachers only teach Reiki and do not give any Reiki treatments. Both have become commonplace, shifting away from the original tradition.
Regardless of the semantics, all Reiki Masters can offer Reiki sessions, and therefore can be referred to as Reiki healers.