Remote diagnostics is the act of diagnosing a given symptom, issue or problem from a distance. Instead of the subject being co-located with the person or system done diagnostics, with remote diagnostics the subjects can be separated by physical distance (e.g., Earth-Moon). Important information is exchanged either through wire or wireless.
When limiting to systems, a general accepted definition is: "To improve reliability of vital or capital-intensive installations and reduce the maintenance costs by avoiding unplanned maintenance, by monitoring the condition of the system remotely." 
Remote diagnostics and maintenance refers to both diagnoses of the fault or faults and taking corrective (maintenance) actions, like changing settings to improve performance or prevent problems like breakdown, wear and tear. RDM can replace manpower at location by experts on a central location, in order to save manpower or prevent hazardous situations (space for instance). Increasing globalisation and more and more complicated machinery and software, also creates the wish to remote engineering, so travel over growing distances of experienced and expensive engineering personnel is limited.