Match Day is the day in the Main Residency Match when applicants learn which program they matched with. It comes at the end of Match Week, during which applicants first learn whether they matched at all, and those who did not may take part in SOAP.
The NRMP runs the Match using the rank lists submitted by applicants and programs (see rank order list). Match results are binding commitments for both sides.
The NRMP publishes the exact dates each year, so check its official site rather than a blog or forum. For what happens day by day, see Match Week.