"right traffic pattern" vs. "right-hand traffic pattern"
"Right traffic pattern" and "right hand traffic pattern" are one in the same.  Usually on the radio either a controller or a pilot will abbreviate it to simply "right traffic" to expedite radio communications.  

For example, A C-152 student pilot is inbound for a controlled airport and says something like "Plainview Tower, Cessna one two five X-ray ten miles to the south at two thousand, inbound with ATIS Golf."  If a controller wants a student pilot to fly a right traffic pattern to land on runway 16 which is active, they might say something like "Cessna one two five X-ray enter right downwind runway 16.  Report abeam the tower."  The pilot then responds reading back the instructions to the controller and might say "right downwind one six.  Report abeam.  Two five X-ray."