Reflexology is an ancient (ca. 3000 BC) Chinese and Indian system for diagnosis and treatment of many conditions.
The theory behind reflexology is that areas on the feet, mainly on the sole but also on the dorsum, correspond to areas and organs of the body. A reflexology practitioner assesses the patient by palpating the feet to find areas that feel 'gritty', which correspond to areas of the body which are having problems.
By manipulating the areas on the foot, reflexologists believe they can affect the corresponding body parts or organs. This is explained as being similar to the phenomenon of 'referred pain'; i.e. pain from the gall bladder is felt in the right shoulder, due to shared innervation. However, there are no anatomical correlations to support reflexology; no nerves run from the reflexology areas to the organs they are said to affect, and the acupuncture meridians for many of them do not run through the feet.
While it cannot be explained, there is a fair amount of evidence to support that it does, somehow, affect them in some way.