It covers you for the whole month, so starts the 1st day of the month. You can apply from 3 months before you turn 65 until 3 months after. If you sign up before your birthday month, it will start day 1 of your birthday month. If you sign up during your birthday month or in the 3 months after, it will start day 1 of the following month.
Note: Part A that covers hospitalization, is paid for by premiums you’ve been paying your whole life. If you don’t sign up in those 7 months when you’re initially eligible, there is a lifelong penalty. This applies even if you have employer provided health insurance. There is no charge/ premium for Part A once you are eligible. So it’s best to start at least Part A when you turn 65, to avoid the life long penalty.