Reiki is much more than a complementary healing therapy; it is a complete self-healing system which can also be used to heal others.  Reiki is extremely popular, especially 'Reiki Level 1' which can easily be learned in a couple of days but can then be used immediately to self-heal and to heal family and friends.  In 2001, it was estimated that there were at least half a million people in the UK attuned to 'Reiki Level 1' and numbers were increasing rapidly; now, nobody knows.

The word 'REIKI' is a combination of the Japanese Kanji: 'REI' and 'KI'.

REI means 'Universal' but in more than just the physical sense.  According to Shingon Buddhism, a being called 'Dainichi Nyorai' created the Universe, everything in it, and also is the Universe — he is literally 'everything' and 'everywhere', he is the sum totality of all that is.  REI represents 'Dainichi Nyorai' thus REI represents the Universe and the God-Force that created it and maintains it.
KI represents the life-force that pervades the Universe and flows through all living things (it is called CHI in China and PRANA in India).  The Japanese dictionary defines KI as ‘vital energy connected with the breath’; ‘invisible life-force’; and ‘material force of the universe’.

Combining the two Kanji to give REIKI also gives a simple definition of Reiki:

Universal or Spiritual Life-force

Reiki works on all levels — mental, physical, emotional and spiritual — to balance, energise and heal both body and mind.  Research studies have shown it to promote wound healing, lower blood pressure, reduce stress, ease chronic pain and also acute pain such as that suffered by people who have been badly burned, boost the immune system, and also reduce the side-effects of medical treatments such as chemotherapy.  It has been used successfully at a Cancer Clinic in Canada to manage pain in patients with terminal cancer giving them a better quality of life in their last days and reducing the need for opiates.  There are also anecdotal reports of it curing serious illness.

Visually, Reiki is very similar to Spiritual Healing; they are both energy therapies using an energy-model of the human body rather than the orthodox mechanical model of western medicine; and the outcome is the same (i.e. a healer channels energy into a patient to bring about a beneficial effect).  The main differences lie in how that outcome is reached:
Reiki is passed on from Master to Student via an Attunement process that looks similar to Shamanic Healing rituals; the Attunement process allows a permanent connection with the Energy without the need for further self-attunement.  This differs from Spiritual Healing where the student is taught to self-attune to the Energy by more experienced healers.
Reiki has three levels of attainment: those who have attained 'Level 1' can self-heal and heal family and friends; those who have attained 'Level 2' are classed as Reiki Practitioners and are qualified to heal members of the general public; those who have attained 'Level 3' or Master Level are qualified to attune and teach others.  This differs from Spiritual Healing which only has two levels (i.e. Trainee Healer and Full Healer).
The historical background of the two therapies is very different.  Spiritual Healing is from the west and has developed over thousands of years from both Christian and Pagan healing rituals.  Reiki on the other hand is oriental; it was developed in Japan by Mikao Usui in the late nineteenth or early twentieth century; it has a lot of similarities with healing and cleansing rituals from Shintoism, Shamanism and esoteric Buddhism which has led many to conclude that it is a re-discovered ancient healing system.  Reiki also differs from Spiritual Healing in that there is much more of a focus on self-healing.
Reiki History

Like Spiritual Healing, Reiki works equally well on plants and animals; it is not 'faith healing'; it is not a religion; and neither the healer nor the patient needs to have any particular religious belief to use it or benefit from it.  Whether used on self or others, Reiki accelerates the healing process; it is very gentle but also very powerful.