Residency applications for positions in the US open in September of fourth year. The electronic system for managing application material electronically is ERAS (Electronic residency application service.) Interviews happen November to mid January. Out of the list of places where you were interviewed, submit your ranked list of preferred residency locations via the US National Residency Matching Program (NRMP). Results of the match are released in March of fourth year.

How many applications do you need to make? American MSIH students need about 100, while international MSIH students may need 150-250. Residency matching services like Match A Resident may help you streamline this number with a list of places they think you're qualified/likely to match.

November - register to NRMP
January - NRMP rank order list opens
February - NRMP rank order list closes
March - Match results announced

Pro tip: If you need a visa to train in the US, it will be a J1 visa sponsored via the ECFMG

Canadian match runs first, so if you match in Canada, you are automatically removed from the US match.

See also
AMA residency search tool: FREIDA - lists program details like compensation, policies, programs, percent of IMGs and number of slots available.
National Resident Match Program