Osteopathic medicine was founded in 1892 by Andrew Taylor Still who received his M.D. degree, but did not particularly like the practice of medicine during his time. Physicians in the late 19th century focused entirely on the symptoms of disease and looked for certain remedies that could alleviate the patients' symptoms. He thought the body could actively repair or heal itself if in correct alignment. His focus was on the musculoskeletal system for the body to maintain proper functioning.
Osteopathic medicine has been extended to Europe, but beyond that I'm not quite sure. Not every country in Europe recognizes the D.O. degree, but many of them do, I believe most in the southwestern countries.
The D.O. and M.D. degree tracks are very similar, they include the first two years as training in the basic sciences and the third and fourth years doing clinical rotations. After graduation, you begin your internships and residencies just as M.D.s do. Osteopathic physicians are eligible for every specialty and subspecialty and licensed to practice in every state in the US. However, the main difference between D.O.s and M.D.s is the manipulative techniques learned by osteopathic physicians and there are a few more residency options for D.O.s, I believe.
Osteopathic Manipulative Medicine (OMM) is a specific modality taught to osteopathic physicians as a diagnostic and treatment technique. Some osteopathic physicians claim it is the most effective tool taught in medical school and others use very little of it in their practice. It is a residency option if that is the route that you choose to go.
I hope this helps you a little bit.