Differences between the gravitational constants $G$ and $g$? There's a formula (described by Sir Isaac Newton) that gives the force acting between two objects: 
$$F = \frac{Gm_1m_2}{r^2}$$
And then there's a formula for weight of an object
$$w = mg$$
My question is, what's the difference between $g$ and $G$ (or force of gravity, just gravity and  acceleration due to gravity). Analogies would help! 
 A: Your weight $w$ on the surface of the Earth is the force $F$ that the Earth exerts on you. So, your weight is $w = F = \frac{G m_{Earth} m_{you}}{R^2}$, where $R$ is the radius of the Earth. To calculate someone else's weight, you'd have to replace the $m_{you}$ with their mass and repeat the calculation. You may notice that $\frac{G m_{Earth}}{R^2}$ remains constant. You can call that $g$ and it evaluates to about $9.8 \frac{meters}{sec^2}$.
It is also called the acceleration due to gravity because from $a = \frac{F}{m}$, using the above expression for $F$ and $m_{you}$ for mass, you are left with $g$. So, in free-fall, this is the rate at which you are accelerating close to the surface of the Earth.
A: For one thing, they have different units. $g$ has units of acceleration while $G$ has units of $(length)^3 / mass / (time)^2$. Physically, $g$ really is the acceleration of a falling object. Usually this is specifically for an object falling toward the Earth (and near the Earth's surface), and $g \sim 9.8 m/s^2$. In fact,
\begin{equation}
g = {G m_E \over R_E^2}
\end{equation} 
where $m_E$ and $R_E$ are the Earth's mass and radius. So you see that $g$ is really calculated from Newton's law of gravity applied to objects near the Earth's surface (so that $r$ is approximately $R_E$).
I think of $G$ as more or less just a numerical and unit constant. If we used different units for mass or length, then there wouldn't even be a need for $G$. 
A: To connect the two you need to know: Outside of a (homogenous) sphere the gravitational force on a massive body is the same as that of a point mass, located in the center of the sphere. So to good approximation the force of the Earth exerted on a body of mass $m$ is
$$F = \frac{G m_{Earth}m}{R^2}$$
where $R$ is some approximate value of the earth radius, the prefactor of the mass $m$ is summarized in the constant you call $g$. Gravity on earth is discussed in more detail on wikipedia.
