A soccer ball curves when you kick it because the ball itself is spinning. By spinning, the ball unevenly deflects the air that’s traveling around it. To balance that deflected air, the soccer ball will experience a force that is perpendicular to the way it’s moving. It is this perpendicular force that pushes the ball to make it curve.

This phenomenon is generally called the Magnus Effect, and it shows up in other sports, too! A couple examples I can think of are putting topspin on a tennis serve, or pitching a curveball in baseball.