Although Newton's theory of gravity has been overthrown by Einstein's Relativity, It still holds true that one can make accurate predictions about the motion of objects using Newtonian Mechanics. For instance, things such as the acceleration of gravity (on earth) is derived from Newtonian Gravity, and is very accurate. It would probably be a very good Idea to look at the theory before moving on to more advance things. (especially if you are like me and your math skills reach their limits past Newtonian Mechanics.) I like the equations from Newton's gravity because you can understand them with just basic algebra, geometry and science behind you pretty easily. Furthermore these equations have been used to understand one of the most fundamental forces for hundreds of years, and is a very useful thing to know regardless.
Step 1: The Equation
The equation that is used to find the gravitational force between two objects (typically one very massive and one less so) is pretty simple, and some what elegant to me. F= (mMG) /D2 Let's begin to dissect this. F stands for Force, or more specifically the force of gravity. m stands for one of the masses involved in the gravitational interactions typically the less massive one. M stands for the other object involved in the gravitational interaction, typically the more massive one. D would be the distance between the objects. G is a particularly interesting part of this equation (equally essential). You may have noticed that it seems small things are not attracted to small things. Certainly you do not have less massive objects circling you right now. This is where the variable G or the Gravitational Constant comes into play. if you were to exclude it from the equation you would notice that any two things with mass would attract each other with a good amount of force even from macro distances. Obviously this doesn't happen. the attraction between two objects that are not as massive as planets is not even enough to really notice (none the less is there, but is just very tiny).
Go to the next step for the gravitational constant.
Step 2: The Gravitational Constant
Newton Knew that there had to be a constant that the masses of the objects were multiplied by. This such constant was never found in Newton's lifetime, but is known now. It to me shows just how weak gravity is compared to the rest of the forces such as electromagnetism. The constant is an incredibly small number. It is about 6.7 (10-11). That is an incredibly tiny number, which means that if you had two objects a meter apart weighing a Kg each, there would be no noticeable gravitational force between them. However is one or both of the objects involved has a very large mass, you will begin to get a noticeable affect.
Step 3: Density?
Using the equation, you may be wondering how it could be used to find the gravitational interactions of things on the surface of a large object. At first glance it may appear that you would put zero for the distance between the two objects. It is not very difficult to see the problem with this though. Dividing by zero is going to lead to many problems. But using this you do not need to worry about the dangers of dividing by zero, because newton represented all of the mass of the object in this equation at the center of the object (It should be mentioned the center is a single point). This would mean, if m (referring to "little m") were on the surface of M, you would plug the radius of the object into the equation for the distance. Going off of this, it can be noticed that if you were to simply reduce the radius of the object without reducing the mass (making it more dense) the gravitational interaction becomes stronger. What happens if the object is nothing more than the single point which should be the center, or an infinitely dense point? Well relativity tells us that you would get a black hole, and it can almost be inferred from this, as gravitational interactions become stronger for things on the surface of an object as that object becomes more dense, that a black hole would occur, but the equation itself is not capable of directly showing the possibility of its existence. Besides the affect of density on gravity only holds true for objects on the surface.
Step 4: Acceleration?
We all know that newton earlier established that F=ma. So then where is the acceleration in our new equation? It is quite simple. simply the same equation excluding m (little "m"). This is how we get the gravitational acceleration on earth being 9.8 m/s2
This number is very important in many calculations here on earth involving trajectories, rocketry, and so on. And of course it can be be derived from Newtonian Gravity.