How does the law of universal gravitation apply to planet motion around the Sun?
The Sun's gravitational force pulls on the planets which move perpendicular to this force at all times creating a elliptical (ideally circular) path around the Sun.
Let's use the Earth as an example:
As the Sun pulls the Earth towards it, the Earth moves very quickly (velocity around 30km/s) perpendicular to the force of gravity. These two forces create a resultant force pointed between the Sun and the Earth's path (let's just say this 30 degrees). Therefore, at all points around the Sun, the Earth will move at a 30 degree angle, creating a circle with radius r (r=93 million miles).