It occurs to me that most people outside of medicine don't really know how the medical education system works after medical school. So here is a combined primer/update.
Upon completing medical school, you are an MD, however you cannot practice clinical medicine without supervision until you complete a minimum of one post-graduate year. Typically, this is accomplished by entering into a residency program. The vast majority of people complete a full residency, upon completion of which you sit the board exams in a particular specialty (or specialties). If you pass that exam, you are then "board-licensed" in that area of medicine. You can further specialize within that category of medicine by completing a fellowship and sitting sub-specialty boards.
Example: You could complete an internal medicine residency to become a board-licensed internist and subsequently complete an endocrine fellowship to become a licensed endocrinologist.
You cannot complete a fellowship without first completing a residency. Theoretically you can practice medicine without board licensure, however you would never get the malpractice insurance required and no hospital would hire you/give you privileges.
Okay, so how does a medical school graduate get into a residency program? The answer is the national residency match process. In your final year of medical school you apply to residency programs at a variety of hospitals using a central database called the ERAS. You fill the application out once and select the programs at which you would like to be considered (and pay $$). Generally, people apply to only one type of residency (ie anesthesia, pediatrics, ob/gyn), however occasionally people will apply to a less competitive option in case they do not get a spot in something coveted (ie, someone applying for dermatology may also apply to internal medicine).
Programs that found your application compelling will invite you to interview with them. An interview involves flying out to the hospital the day prior, attending a dinner with current residents the evening prior, followed by a full day of interviews/presentations/tours. You pay for this yourself, so you can imagine that a person who interviews with 10-12 programs will burn though a chunk of change accomplishing this. Interview season is November-January.
In February, the applicants submit a ranked list of places they interviewed at. The list does not have to include all the programs, however, not ranking a program means you would refuse to work there even if given an offer. Applicants cannot rank programs at which they did not interview. Simultaneously, each program ranks the applicants they interviewed. Again, they do not have to list all the people they interviewed, but they cannot list someone who did not interview.
In March a computer goes through and attempts to create the optimal pairing of applicants to programs. If multiple programs rank an applicant, the applicant is generally assigned to the program they preferred. If no programs rank an applicant high enough, that applicant will not get a residency placement (ie they will not have a job). On March 16th, 2012, all applicants will get an email telling them where they have been assigned. If an applicant did not match, they are generally notified the Monday prior and they go through something called the scramble.
So, where am I in all of this? Well, I'm applying to a combined residency program, called medicine-pediatrics. It's four years in length. At it's conclusion, I will sit both the internal medicine and pediatrics board examinations (ie I will be double-boarded). I applied to 19 programs and was offered interviews at 14 of them. I completed 11 interviews, but have not yet settled on what order to rank them.
Places I interviewed: UPenn/CHOP, Univ. Minnesota, Univ. Maryland, Univ. Michigan, Georgetown, Brown, Baystate (Tufts), UCLA, USC, UNC, Univ. Rochester
Places I declined to interview: UCSD, Vanderbilt, Univ. Chicago
Places I was rejected: Brigham & Women's, Mass Gen, Yale, Duke
Places I never heard anything from: Univ. Pittsburgh