Use the phrase to express that an event is extremely unlikely to occur. He's so stubborn, he won't apologize in a month of Sundays.
Pro Tip 2/3
Exaggeration
The phrase is often used for hyperbolic effect, not to be taken literally. You could wait in a month of Sundays, but you still wouldn't see a bus here.
Pro Tip 3/3
Humorous Effect
Sometimes it's employed humorously to highlight the extent of a delay or inaction. I'll grow old and grey waiting for her in a month of Sundays!