It depends on the program. In the US, most full-time MBA programs will start in the fall. However, there are some different options. For example, Kellogg has a one-year MBA program that starts in July, and Columbia has their J-term that starts in January.
For part-time programs, I believe at a lot of schools you apply to start in a specific term/quarter, so you will have students starting at different times throughout the year.
The actual date will vary from year to year but not by much. Mostly it varies so that the program will start on the same day of the week each year. You should be able to find the school's academic calendar, which shows the exact dates, on their website.