@liz and @Dragon,
say you have a tee-time scheduled for 2:00pm this afteroon, but it starts to rain (even lightly, not a real summer shower): you can "take a rain check", so that you can play some other time in the future, but not today at 2:00pm (in the rain). In other words, although you are not going to be playing today, you do not want your fees to go wasted, and you hang on to the chance of playing some other time.
Taking a rain check expresses a little bit of regret on the side of the person who turns down the invitation (I wish I could, but I can't), as it also expresses the desire to accept the same type of invitation some other time in the future. In other words: "I wish I could, but I can't. Nonetheless, please invite me again, if you're doing the same thing some other time in the future!! I will gladly come along!!"