Spicy things have a sharp, strong, pungent flavor. Hot chili peppers—and food that's made with them—are notoriously spicy.
People who don't like spicy food might be hesitant to try an Indian, Thai, or Mexican restaurant, since the cuisines of those countries are known for being spicy. But if you love the way curries tingle on your tongue, then you're a fan of spicy dishes. Spicy is also used to mean "exciting but vulgar or inappropriate," like a spicy joke that's not suitable for school.