If you go to a waterpark and it starts to rain, they will usually give you a rain check. It's a free pass to come again at a later time (when it's not raining).
"This term comes from baseball, where in the 1880s it became the practice to offer paying spectators a rain check entitling them to future admission for a game that was postponed or ended early owing to bad weather." -urbandictionary.com
Also, if you go to a store and want to buy a sale item that is out of stock, you can request a rain check to buy it in the future for the same price.
In everyday English, it is something that you say when you cannot accept someone's invitation to do something but you would like to do it another time. For example:
I can't go to the movie tonight, but I'll take a rain check, if that's all right.
I won't play tennis this afternoon but can I take a rain check?
I can't come to dinner Tuesday but I hope you'll give me a rain check.