A lot of good information is available on the Navys of the world in general and the U.S. specifically here on Everything2. I will focus on some of the organization here, to cut on redundancy.

"In the Navy"

Mission Statement of the Navy

The mission of the Navy is to maintain, train and equip combat-ready Naval forces capable of winning wars, deterring aggression and maintaining freedom of the seas.

Foundations

The U.S. Navy was founded on October 13, 1775, and the Department of the Navy was April 30, 1798.

High Level Organization of the U.S. Navy

             Secretary of the Navy
                        |
                        |
       -----------------------------------
       |                                 |
Chief of Naval Operations          Commandant of the Marine Corps
           |                                      |
           |                            Marine Operating Forces
           |--------------------                .
           |                   |         mutual support
Naval Shore Establishment      |         .   
                         Naval Operating Forces

During wartime, the U.S. Coast Guard becomes part of the Navy, during peacetime, it is part of the Department of Transportation.

You may see the Marines listed up there because the Marines get their paychecks courtesy of The Department of the Navy. This may be part of the cause of some animosity between the branches of the military.

Naval Operations Jurisdiction

Once you get down the ranks a little bit, you come across the Chief of Naval Operations or the CNO. The CNO is a Four Star Admiral. Under the CNO is a massive bureaucracy known as the Naval Shore Establishment.

Under the CNO, in the Naval Operating Forces department, is the Commander in Chief U.S. Pacific Fleet and Commander in Chief U.S. Atlantic Fleet. Under these are split into three parts per Ocean: Submarine Command, Surface Fleet, and Airborn.

Rank in the U.S. Navy.

Enlisted

  1. Seaman Recruit
  2. Seaman Apprentice
  3. Seaman
  4. Petty Officer Third Class
  5. Petty Officer Second Class
  6. Petty Officer First Class
  7. Chief Petty Officer
  8. Senior Chief Petty Officer
  9. Master Chief Petty Officer
  10. Master Chief Petty Officer of the Navy

There are four levels of Warrant Officers.

  1. Warrant Officer (W-1) (Obsolete)
  2. Chief Warrant Officer (W-2)
  3. Chief Warrant Officer (W-3)
  4. Chief Warrant Officer (W-4)

Officers

  1. Ensign
  2. Lieutenant, Junior Grade
  3. Lieutenant
  4. Lieutenant Commander
  5. Commander
  6. Captain
  7. Rear Admiral, Lower Half (One Star)
  8. Rear Admiral, Upper Half (Two Stars)
  9. Vice Admiral (Three Stars)
  10. Admiral (Four Stars)
  11. Fleet Admiral (Five Stars)

Random Bit

A good friend of mine is a Gulf War Veteran, he worked on an Aircraft Carrier on the Flight Deck. He says the only difference for the Flight Deck Crew between war and peace is that the planes come back without the bombs.

My American Government professor told me the most explicit example of the Peter Principle was in effect during his tenure in the Navy. His CO was a wonderful Mechanic, but, once promoted, could not function in his new role. I.E., he was a bad boss, and knew it, too, and requested to be "demoted," actually transferred, to a similar position befitting his promoted rank.

Source

A lot of this info is taken from the Navy web site (http://www.navy.mil).