What is the meaning of “American 755, turn right, heading 320"?
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.