What is the meaning of the density of a distribution at a point? I know how to use PDFs to calculate probabilities, but I don't think I understand them. For example, at $X=0$ the PDF of the standard normal distribution is $\approx 0.4$. Does this have any useful meaning?
 A: $f(0)$ is the density at 0. 
It is meaningful in several ways. 
For example, the probability of being within a small distance of $x$ ($\pm \varepsilon/2$) of $x$ is approximately $\varepsilon f(x)$.
It is relative probability; for the standard normal $f(0)$ is $\sqrt{e}f(1)$, so a value very close to 0 is, relatively speaking, 1.65 times as likely as a value that close to 1.
A: Also note that while the density at a point has a value, the probability at a point for a continuous distribution will always be zero as the area under a point is 0.
A: As you know, a probability density is not a probability.  One interpretation of density considers the relationship $$f_X(x) = F'_X(x).$$  In this context,the density at some value $X = x$ is the instantaneous rate of change of the cumulative distribution; i.e., how rapidly the probability of observing $X \le x$ is increasing.
Another interpretation comes from the limit $$f_X(x) = \lim_{\Delta x \to 0} \frac{1}{\Delta x} \Pr[x \le X \le x + \Delta x].$$  In this sense, the density is the differential probability of observing $X \in [x, x + \Delta x]$ divided by the length of the interval $\Delta x$.  So in some sense it represents a likelihood of observing $X \in [x, x + \Delta x]$, with larger densities reflecting a larger likelihood of observing values in that interval.
A: Before answering your question directly, one important thing to note is that, for continuous variables, the density at $X = x$ cannot be interpreted as the probability that $X = x$. Indeed, the density at any given $x$ could be greater than one because all that matters is that the density integrates to one, and the intervals are infinitesimally small.
With that background in mind, the density does have several useful meanings. One is that it can be used to compute your relative belief that $X = x_1$ versus some other $x_2$. To do this, simply take the ratio of the two densities.
So although we are usually more interested in areas under the curve with width greater than zero, areas under the curve with infinitesimally small width can be compared. The relevance of that comparison, however depends on your research question.
To give you a concrete example of when comparing densities is useful, I point you to a question I recently asked on Cross Validated about useful prior distributions for a correlation coefficient when you want to avoid the boundaries of the distribution. I argued that one possible prior distribution, a beta distribution with both parameters equal to two, is quite informative in that it places about seven times the belief in the correlation being zero than in it being moderately negative at about -0.4 or strongly positive at about 0.94. To do that, I divided the approximate density at $x = 0$ by the approximate densities at the points in the domain of the beta distribution (which is 0,1) and got the number seven. So the Beta(2,2) distribution has seven times stronger of a belief that the correlation is zero than that it is moderately negative or strongly positive.
I hope this helps.
