How do you get accepted into a Military Academy?

Acceptance into one of the five federal military academies is extremely competitive, and even the most qualified candidates may find it difficult to gain an appointment. Prospective candidates must have high class ranks, grade point averages, SAT or ACT scores, and must excel in athletics and participation in clubs and activities. Students need to show leadership potential, have good conduct records, and pass a physical fitness test, medical evaluation, and interview. In addition, they need to be nominated by an appropriate authority, namely their congressman or one of their senators. Each academy's admission requirements may differ slightly, but share these basic mandates.
Each academy is a four-year, federally-funded accredited university. Graduates receive Bachelor degrees in their majors, as well as a commission into one of the branches of the armed forces. With graduation they incur a service obligation that requires them to serve in the military in exchange for the education they received.
The primary mission of the federal service academies is to train competent and qualified officers for the U.S. military. The U.S. Military Academy, U.S. Naval Academy, and U.S. Air Force Academy are all operated by the Department of Defense and commission graduates into the Army, Navy, and Air Force, respectively. The U.S. Coast Guard Academy is operated by the Department of Homeland Security and commissions graduates into the Coast Guard. The U.S. Merchant Marine Academy is operated by the Department of Transportation and commissions the majority of graduates into the U.S. Navy Reserve, with approximately 1/4 of graduates being commissioned into the other branches of service in both active and reserve capacities.
More information about the federal service academies can be found at their various websites.