What gives rise to the normal distribution? I'd like to know if anyone has a generally friendly explanation of why the normal distribution is an attractor of so many observed behaviors in their eventuality. I have a degree in math if you want to get technical, but I'd like to be able to explain to my grandma as well

When you plot the computed probabilities of getting $k$, $\ 0\leq k\leq 10$, heads in $n=10$ throws of a coin you obtain a histogram showing a very crude approximation of a bell curve. When you do the same thing for $n=100$ or $n=1000$ throws you get finer histograms of a similar shape, but the peak in the middle gets definitely sharper as $n$ increases. We see here the law of large numbers at work. Doing the calculations one finds that the relative width of the peak decreases proportionally to ${1\over\sqrt{n}}$. Compensating for this effect using suitable scalings one indeed obtains a unique limiting curve, the so called Gauss curve.
Now for the universality of this curve. When doing our histograms, a tail counted $0$ and a head $1$. So the result of the $i$th throw can be viewed as a random variable $X_i$ taking the values $0$ or $1$ each with probability ${1\over2}$. What the histogram shows is nothing else but the distribution of the sum $S:=X_1+X_2+\ldots+X_n$. (This might be a little difficult to explain to your grandma.) Up to a horizontal translation and a scaling the histograms would look exactly the same if the variables $X_i$ assumed the values $\pm1$ or $\pm0.001$ instead of $0$ and $1$.
Imagine now a physical experiment, say the measurement of the temperature $T$ at Kennedy airport on 21.12.2012 at 12:00 p.m., which in our long time experience should turn out the value $0$ ($0^\circ$ C in the case of our example). In reality the exact reading of $T$ is determined by a thousand tiny influences which act independently of each other;, but  each of them is contributing $\pm0.001^\circ$ to the resulting value of $T$. Imagine now that we are doing this experiment for 50 years in a row and make a histogram of the measured $T$-values. We shall see our bell curve again.
