Here's the way it usually works. Match is usually sometime around March 19. 93% of US seniors matched this March. On eg the 16th, three days earlier, students who didn't match were told that they didn't. Then on March 17-18 there is what is known as the "scramble". What that entails is that on March 17, the NBME releases a list of all the programs that didn't fill and starting at noon that day, the students, usually with the help of faculty and deans, furiously make calls and faxes to programs with open slots hoping to snag one of those remaining places. A good percentage of the US students who didn't match get a spot in this way -- the number of residency slots currently outnumbers US applicants -- they just don't always get the specialty they desire or may have to take a one year prelim slot. Non US med students tend to do much worse in the scramble -- it's harder to get through if you don't have a faculty member helping you and able to use a private line or get someone to take their call. So many people reportedly simply spend two days on voicemail and faxing stuff with no results.
The remainder of folks who don't snag a position usually line up a research year to improve their application for next year, or otherwise try to obtain a spot that opens up later during the year (people sometimes drop out due to pregnancy, family issues and folks jump in and snag their spots.)