The Universal Law of Gravitation

Gravity is described from the point of view of a universal law.  This implies that gravity is a force that should behave in similar ways regardless of where you are in the universe.

Gravity
It's a force of attraction that exists between any two objects that have mass.  The more mass they have, the greater the force of attraction.  The closer they are, the greater the force of attraction.

For most objects you get near every day, the force of attraction is so incredibly small that you would never notice the force.  Gravity is a very weak force, so between common objects like you and your pencil, the force of attraction is very small because your mass and the mass of your pencil are small.  We only get noticeable amounts of gravity when at least one object is very massive... like a planet.   The force of attraction between you and the planet Earth is a noticeable force! We call the force of attraction between you and the Earth, your weight.  Weight is another name for the force of gravity pulling down on you or anything else.

 G is the universal gravitational constant.  It is basically a conversion factor to adjust the number and units so they come out to the correct value.  This is a universal constant so it is true everywhere. m1 is the mass of one of the objects. m2 is the mass of the other object. r is the radius of separation between the center of masses of each object. FG is the force of attraction between the two objects.
Important Concepts
The direction of the force is not given by this formula since there are actually two forces equal in size but opposite in direction.  This formula calculates them both.
The formula is an inverse square law for radius of separation (notice the r2 on the bottom of the equation).  This means that if you double the separation you quarter the force, or if you cut the separation in half you quadruple the force of attraction.
If you double a single mass, you double the force.  If you cut one of the masses in half, you cut the force in half.  But if you double both masses you would quadruple the force.

Common Misconceptions
"There is no gravity in space."  FALSE  If there were no gravity in space, the space shuttle would not be able to orbit the Earth, the moon would not orbit the Earth, and the Earth would not orbit the Sun.  The reason we tend to think of there being no gravity in space is that we have seen movies of the astronauts being "weightless".  They aren't actually weightless, they are still being pulled down by gravity but they and the space shuttle are in a constant state of freefall around the Earth.  So they seem to be weightless as a result of the falling - just as you would seem weightless if you were in an elevator when the cable broke.
"G and g are the same thing."  FALSE   Big G is the universal gravitational constant.  Little g is the acceleration due to the force of gravity and its value of 9.8m/s/s down is only true on this planet.  It is not a universal constant.
"g is gravity."  FALSE   Little g is the effect of the force of gravity, but is not gravity.  Gravity is a force, little g is an acceleration caused by gravity.