Someone who's sulky is gloomy, or quietly unhappy. A sulky teenager is generally not a lot of fun at a family party.
Some people tend to be sulky when they're unhappy or disappointed — when you're sulky, you're not just sad, but you mope and frown and sigh. You might be sulky if your brother gets to travel to England and you don't, or if your coworker gets a promotion instead of you. Sulky probably comes from an Old English word, asolcen, "idle, lazy, or slow," which has a Proto-Germanic root.