Even in the Most Magical Place on Earth, it has to rain every now and then. Actually, since it's located in Florida, there often is rain. The good news is that Disney World is pretty well-equipped to deal with rain, having plenty of things to see and do.
Magic Kingdom
Honestly, one of the best things you can do at Disney World when it rains is to put on a poncho and go ride some rides. The lines are incredibly short when it rains, as crowds tend to scatter for indoor fare. At Magic Kingdom, Splash Mountain is typically pretty empty in a deluge, which is obvious on one hand and silly on the other—guests were planning to get wet anyway; why is rain stopping them? Walt Disney World actually has more indoor rides than its West Coast forebear for this exact reason. If you're not feeling adventurous, you can always go shopping. Travel from store to store and nearly reach the park's entrance from Cinderella's castle without stepping outside once. Unfortunately, you're going to be contending with some pretty wild crowds. Another option is to go check out an indoor show like the Tiki Room or Mickey's Philharmagic.
Disney's Hollywood Studios
Disney's Hollywood Studios is all about movie magic, which means there's plenty to do indoors. Hollywood Boulevard has The Great Movie Ride, or you can check out the Star Wars-themed Star Tours in Echo Lake and a 3D Muppets show in Streets of America. Pixar Place also has an indoor ride, Toy Story Midway Mania. It's an on-rails ride where the cars are equipped with toy cannons, used to play several games of accuracy with simulated projectiles and targets.
Epcot
Epcot's not a bad place to get caught in a rainstorm at all. Each one of the country pavilions has some kind of indoor attraction. As with the Magic Kingdom, the shops are going to fill pretty quickly, but there are also restaurants, rides, shows, and movies that'll keep you entertained and dry at the same time. Check out the Reflections of China film or several live entertainers over at the Italy Pavilion. Things that you should not do include climbing the pyramid at the Mexico Pavilion, because while that isn't a good idea to begin with, it's an especially bad idea in the pouring rain.
Animal Kingdom
Like the other parks, Animal Kingdom has no shortage of things to do indoors. Presumably all of the animal-based attractions will be spotty, but the half-hour Festival of the Lion King show in the Africa region is a solid bet for entertainment. Nemo The Musical, over in DinoLand, is just as captivating. Both are large-scale affairs with colorful sets, live performers in elaborate costumes, and intricate puppetry. It's sure to keep the whole family entertained while you wait for the rain to clear up.