After medical school, you are an M.D. (or a D.O. if you go to osteopathy school). You are a doctor, but you still need to get a license to practice. You basically will need to then do at least 3 additional years in a residency in order to practice independently (you technically can after 1 year of residency/internship I believe - but not many people who do that).

In order to become "board eligible" in psychiatry, you would need to do a 4 year residency. It is essentially a clinical experience where you spend your time working in various parts of the hospital as a psychiatrist - learning how to do the things you'd have to do when in a practice of your own. You do get paid a relatively small salary during residency. Upon completion of a residency, you are "board eligible". Once you take and pass the board exams, you are then "board certified" in psychiatry. Board certification is unnecessary for most jobs.