The thing that is gripped in one hand that you strike a drum with. Usually made of wood, sometimes of plastic or some composite material. Most sticks have a shaft of a constant diameter that you grip, and then a shoulder that narrows down to the tip or head, which is pretty small but swells out in diameter again. Anything that's used to strike any percussion instruments other than drums is usually called a mallet.
Most percussionists will twirl their sticks around their fingers if they're sitting back in the percussion section being bored, which happens a lot in rehearsals. Non-percussionists like to play with drum sticks, but this behavior should not be encouraged. It's the equivalent of playing with someone's mouthpiece. Beat them upside the head, or on the fingers if you are in a particularly sadistic mood, to discourage this behavior.