Though there isn't a fixed/official definition of the term, 'Concierge' medicine usually refers to the practice of operating an extremely high priced, cash only practice for a very small number of clients, usually 25-50 total patients on your census. They pay a fixed, extremely high amount of money per year to have 24/7 access to a personal physician.
Its basically impossible to practice concierge medicine ethically. When people pay an extremely high amount for a personal physician they expect you to cater to their wants, not their needs. Think about Michael Jackson's doctor prescribing Propofol as a sleep aid for an extreme example. The ones I have seen were miles away from practicing evidence based medicine. When you have 25 high paying clients you don't give them lectures about diet and exercise, you give them whatever pills that they want and whatever diagnoses their lawyers deem that they need. Not that a concierge doctor would recognize a serious problem if he saw it, of course. Attendings still need to see a lot of patients or their skills rapidly degrade, and concierge doctor see the equivalent of less than 1/2 day of clinic per week.
This is contrast to 'direct primary care', or DPC. Again, there isn't a fixed/official definition of the term, but DPC usually refers to a more moderately priced, cash only practice for a census of 400-800 patients. Again, they pay a fixed, annual sum for access to your clinic and you agree to be their physician, but they're paying you significantly less (usually about as much as they pay for a cell phone bill) and you need each individual client much less. DPC usually involves longer than average appointments (30-40 minutes) and the patients have some after hours and weekend access to their physicians, but its overall a normal physician patient relationship where they come to your office, you give sound advice and indicated medications, and if the patient begins to make wildly unreasonable demands they part ways with the practice.