When talking about sport, a rain delay is a delay in a match caused by rain. Some sports are very sensitive to rain and will take a break at every little shower or drizzle (tennis and golf) whereas other sports won't stop for anything less than a full-blown typhoon (soccer and football).
If you're a spectator, a rain delay is annoying because here you are, you've set aside a big chunk of the day to watch something and you can't. If you're a participant, rain delays are equally bothersome because you're mentally "psyched up" for the match and then you have to put it all on hold. Also in a rain delay your body cools down so unless you're careful to warm up again, there's a greater chance of injury.