It means "adverse factors will not effect an event."
"We will have our party (game, meeting, date, or other appointment) tomorrow rain or shine." No matter what the weather is (rain or sunshine), we will have our party.
The expression is also used for events that cannot be affected by weather.
"I'll be in your office tomorrow at 9 AM rain or shine. I won't let anything stop me from getting to your office.
- For Teachers