What is the meaning of “American 755, turn right, heading 320"? I am studying a text entitled Unmanned Aircraft Systems. How can we simplify the instruction in quotes so that it is more understandable and can be understood by a nontechnical. Please note I am not seeking to know all technical jargon of routing aircraft, rather to get what a pilot is told with this instruction to do.   


  Ideally, the communication process includes three parts: the
  initial call, readback, and hearback. For example, an ATCO would issue an instruction “American 755, turn right, heading 320.” Pilots onboard the aircraft would respond, or readback the message “Right, heading 320, American 755.”
American 755, turn right, heading 320.


Aviation phraseology is direct, specific, and concise. The three items here:


The controller is instructing the aircraft identified as American 755
to make a right turn 
such that they maintain a heading (direction) of 320 degrees (5 degrees north of north-west).




Though you may not be interested right now, at some level you might have to get technical, as communication involves defining terms :) 

In case you are interested, for references and further reading see this page from the FAA. Especially of note:


Pilot/Controller Glossary - contains definitions for radio communications.
Aeronautical Information Manual - contains instructions for pilots.
JO 7110.65 - (more technical) the handbook for controllers. Everything they say should be in there.