Hey mmpp that is an excellent article you posted. Thanks.
nurophillic - Kudos for asking a simple but essential question!
For answers to your questionsI would definitely recommend reading the article that mmpp posted. But, you might also want to search out some sources beyond the web. You could start by checking some of the references in the article that mmpp posted.
Read pretty much everything by A. T. Still to learn the basics of osteopathic philosophy.
For the history of osteopathy, Norman Gevitz's "The D.O.'s: Osteopathic Medicine in America" is required reading.
Andrew Weil's "Health and Healing" is a good place to start learning about the similarities and differences of the various major medical systems, such as, allopathy (M.D.'s), osteopathy (D.O.'s), homeopathy (D.Hom.'s), chiropractic (D.C.'s), naturopathy (N.D.'s) and practioners of traditional Chinese medicine (O.M.D.'s and Lic. Acc.'s).
However, the short answer to your question only M.D.'s and D.O.'s have licensure to prescribe medicine and perform surgery in all 50 states. Obviously, the M.D. degree is most recognized and most prestigious. Although D.O.'s can legally qualify for any residency-training program and enter any medical specialty, most are primary care physicians.
Homeopathy has a weird status. Homeopathy is based on the "Law of similars." The theory substances that would normally produce symptoms can cure these same symptoms when given in highly diluted preparations. These preparations are often diluted 10^-200 times, well past Avagardro's limit or what could be detected with scientific laboratory equipment. Yet, the FDA does recognize and regulate the manufacture of homeopathic medicines. There is no conventional scientific explanation for the efficacy of homeopathic medicine. However, there are multiple double blind, controlled studies that demonstrate the efficacy of homeopathy over placebos. Weird stuff.
States differ as to who can practice homeopathic medicine. In the U.S., there are few actual schools homeopathic medicine left and no remaining homeopathic hospitals. However, some prestigious allopathic medical colleges, including Hahnemann and University of Michigan, were once homeopathic schools . Today, many practioners study homeopathy through correspondence schools. Some states allow practioners whose only medical education is through these schools. Many others states require another medical license such as, M.D., D.O., R.N., D.C., etc. to practice homeopathy.
Good luck.