While our sons were deployed to the "Big Sandbox", we sent many packages using the "one Rate" boxes provided by the post office. We found this the best "buy" for our money especially if it was heavy. When our oldest son requested a horseshoe set, you can imagine the weight of shipping those!!
APO is a term referring to an "Army Post Office" or a military post office overseas (i.e. Korea, Germany,etc.). Postal rates are similar to those here stateside. When sending to/from one you have to fill out a form stating contents of packages. if its something you want to surprise the recipiant, you can state "bonafide gift" and value.
As for the problem Joan and fktsewing had with customs and cost, its may have been because it was not a military APO. Not everyone has access to use an APO.
Sorry if this is too long but maybe it can clear up a couple questions. Thanks for indulging my "two cents worth". :-)