To double over is to bend forward, usually because you're in pain. If you suddenly double over in math class, clutching your stomach, you should probably see the school nurse — it might be appendicitis!
When you're in so much discomfort that you bend over, essentially folding your body in two, you double over. This phrase also works for extreme cases of laughing, so if you collapse forward with giggles after hearing your friend's hilarious joke, you can also say you double over. You'll know your career as a stand-up comedian is off to a great start if your audience doubles over with laughter throughout your first set.