To swirl is to turn in circles or spirals, the way a leaf caught in a whirlpool swirls, or a toy duck swirls in the bathtub as you empty the water out.
Smoke might swirl up from your campfire, keeping the mosquitos that swirl through the air nearby from biting you. Or you can swirl on a carnival ride that spins you in circles until you step back onto the ground and the world swirls for a while, until you regain your balance. Swirl is a noun, too, as in a swirl of dust. It was originally Scottish for "whirlpool."