If you hold a piece of string in two different locations around a balloon, the string will wrap around the surface of the balloon in each location. This happens because the string is under tension, and the tension forces it to curve and take the shortest path between the two points where it is being held.
The curvature of the string is due to the fact that the balloon's surface is not flat, but rather curved in three dimensions. The string follows the curve of the balloon's surface because it is the path of least resistance for the tension forces acting on the string.
In general, the curvature of the string will be more pronounced in areas where the surface of the balloon is more curved, and less pronounced in areas where the surface is flatter. If the balloon is perfectly spherical, the string will form a great circle that divides the surface of the balloon into two equal hemispheres.