My answer in two words is "bean machine." I'll end with a little about that. But concerning such a fundamental topic as the normal distribution, it is worthwhile to develop intuition for more than just its shape.
A familiar example of a value that is approximately normally distributed is the IQ test score. The test is defined and periodically recalibrated so that 100 is the average (arithmetic mean) score and 15 is the standard deviation. To convert to z-score, simply subtract 100 from IQ and divide by 15.
Chebyshev's inequality is a rule of thumb that helps to make the standard deviation and the mean more familiar and to place the normal distribution within the context of arbitrary probability distributions. It states that no matter what the distribution, no more than $1/k^2$ of the probability mass (roughly, the number of sample values) are farther than $k$ standard deviations from the mean.
The normal distribution is a bit special in that it is considerably more peaked than the loose estimate from Chebyshev's inequality can predict. Another interesting property is that the normal distribution changes from curving over the body of the bell to curving under the flare of the bell exactly at the standard deviation.
A much more special property of the normal distribution is its place in the central limit theorem. Roughly, the central limit theorem states that, no matter what distribution we sample, the mean of the sample is an approximately normally distributed estimator for the distribution's mean. Wherever we take an interest in averages, we will encounter the normal distribution.
The z-score is the standard way to express where a value lies in relation to a mean and standard deviation--often the standard deviation of observations of the mean itself.
A principle that opens clearer graphical and mathematical intuition is that the normal distribution is the limit of the binomial distribution as parameter $n$ increases without bound. The binomial distribution in turn is what we get from adding many Bernoulli trials (such as coin tosses).
Upon those principles Francis Galton designed the "bean machine" device. From the top center of this inclined box that we view through its open front, beans or balls are dropped through a field of pins with just enough space between the pins. Upon hitting each pin, the falling body produces one Bernoulli trial by dropping to the left or right. The many falling bodies pass over many pins and land on the flat bottom of the device. They produce a sample from a binomial distribution. The number of rows of pins is the number $n$ of Bernoulli trials. That and the number of samples from the resulting binomial distribution are large enough to produce a bell curve that approximates the normal distribution.
You could perhaps obtain a particularly large $n$ by dropping a stream of fine grains of flour from a height so that they would be dispersed by the resistance and Brownian motion of the air.