When the arrow is in the air, the gravitational force acts on its center of gravity, and rotates it about the axis of rotation. The gravitational force in this case provides the centripetal force to the arrow which allows it to curve in a parabolic path.
It is important to note that action of gravitation force is such that the arrow always remains tangent to its flight path.
Some examples of the force of gravity include: The force that holds the gases in the sun. The force that causes a ball you throw in the air to come down again. The force that causes a car to coast downhill even when you aren't stepping on the gas. The force that causes a glass you drop to fall to the floor.
The gravitational force on Earth is equal to the force the Earth exerts on you.
At rest, on or near the surface of the Earth, the gravitational force equals your weight.
Learn more about gravitational force here: