Areas under the graph represent probabilities. That's why the total area under the graph is 1.
On the horizontal axis will be the quantity that has a Normal distribution. For example, if we are investigating the mass of people then the quantity on the horizontal axis is mass in kg.
The normal distribution is a continuous function so the probability of any particular value is 0. For example the probability that the mass is exactly 41 kg is 0. The probability density function isn't 0. You can think of that as being the probability of the mass being close to a particular value, or a measure of how rapidly the cumulative probability is increasing.
Remember that the Normal distribution is only a model. It's often a very good model but clearly not perfect because as you point out it could suggest that a mass could be negative 200. But nearly all of the probability is within 3 standard deviations of the mean so the probability of extreme values is very low indeed.