The term is used to describe different programs, but most would agree that, at a minimum, Therapeutic riding is an adapted riding lesson for persons with disabilities.

Therapeutic riding can focus on sport, education or recreation. In the US, therapeutic riding is often provided by a North American Riding for the Handicapped Association (NARHA) certified riding instructor. This riding instructor has had special training in teaching riding skills to individuals with challenges. Some programs expand this term to include medical treatments by professional physical therapists or occupational therapists combined with these adapted riding techniques and lessons. These treatments would be provided directly by these medically licensed therapists who may also be NARHA trained and certified riding instructors.

Therapeutic riding is an example of an Equine Assisted Activity. NARHA defines Equine Assisted Activity (EAA) as an umbrella term inclusive of the various offerings of NARHA centers designed for people with disabilities or diverse needs (in the US). For example, a NARHA center that offers therapeutic riding, vaulting and equine facilitated learning can say that they offer equine assisted activities. Equine assisted activities are provided in an educational frame of reference where the intent is to teach a horsemanship or riding skill. An activity is therapeutic if a participant derives benefit, shows improvement or feels better once engaged. An activity can be therapeutic without being considered as therapy. In general, EAAs may be described as therapeutic, but they are not therapy or is not considered treatment unless the treatment plan is created and executed by a registered therapist (in the US).

Hippotherapy is a treatment method involving horses as a means of working on physical, occupational, and speech-language goals. Impairments that can be improved with hippotherapy include: impaired balance responses, coordination, communication, or sensorimotor function difficulties; poor postural control; and decreased mobility.

The word hippotherapy is derived from the Greek word hippo which means horse. (The word hippopotamus also comes from the Greek and means River Horse.)

History of Therapeutic use of horses:

Therapeutic use of equestrian mounts was pioneered by the Germans in WW2. They called it Reittherapie (ride therapy, implying horse riding).

In Europe, according to the NARHA web site, Lis Hartel of Denmark was also a leader in the creation of therapeutic riding centers (see also link below):

The achievements of Lis Hartel of Denmark are generally regarded as the impetus for the formation of therapeutic riding centers in Europe. Polio impaired Hartel's mobility but not her spirit. In 1952, she won the silver medal for Grand Prix dressage at the Helsinki Olympics. Medical and equine professionals took notice and soon centers for therapeutic riding sprang up in Europe.

Although hippotherapy and therapeutic riding have some risks, as do any animal-assisted therapies, in accredited centers the risks are minimal and the benefits outweigh those risks. This type of therapy is growing in the US with new programs developing around the nation. Programs that are certified by NARHA, the North American Riding for the Handicapped Association, have undergone rigorous inspections and found to provide a safe and therapeutic environment that meets their strict standards.

See the links section associated with this question for additonal information from NARHA and a NARHA-accredited "Premier Accredited Center" in the US, SIRE Houston's Therapeutic Equestrian Centers, a non-profit organization.