Intuitive meaning of the probability density function at a point I understand how to integrate probability density functions to find probability within a certain range.
However, what I don't understand is what it would mean to set the variable (say $x$ or $y$) to a certain value and evaluate the pdf as you would evaluate any other function.  
I know this happens when dealing with conditional density functions in a multivariable situation. But I'm a little confused as to what this actually signifies.
 A: The velocity of a moving particle is the rate of change of the displacement at that time. 
The probability density of a continuous real valued random variable is the rate of change of the cumulative probability at that point.
A: Here's an example that may convey some intuition.
Suppose we consider how much it rains in a given year in Los Angeles.  This random variable, which we shall call $R$, has a distribution, which is a PDF that we may denote $f_R(r)$.
Suppose, for the sake of argument, that $f_R(15) = 0.1$.  Now what does that mean?  Obviously, it doesn't mean that there's a $0.1$ probability that it will rain exactly $15$ inches.  That probability is zero, since the rainfall will differ from exactly $15$ inches* almost surely.
Instead, what it means is that the probability of it raining about $15$ inches is $0.1$ "per inch."  That is, if we ask how likely it is for the rainfall to be in a given range around $15$ inches, the answer will be approximately $0.1$ times the width of the interval in inches.  Obviously, the answer won't be exactly $0.1$ times the width of the interval, in general, but for intervals that are "small enough," the answer will be "close enough" (in a delta-epsilon kind of way for continuous PDFs).
*Ignoring the atomic nature of rain for the moment.
A: Strictly speaking, at any given point the pdf doesn't mean anything. You can change the value of the pdf at that particular point to be whatever you want, and the distribution of the random variable is the same.
Still, "generically", $f(x)$ is essentially the probability to find the random variable in the interval $(x-\epsilon,x+\epsilon)$, divided by $2 \epsilon$. More precisely, if $X$ is a continuously distributed random variable with PDF $f$ and CDF $F$, then
$$\lim_{\epsilon \to 0^+} \frac{F(x+\epsilon)-F(x-\epsilon)}{2\epsilon} = f(x)$$
at almost every point $x$, in the sense of Lebesgue measure. This is one case of the Lebesgue differentiation theorem. The "exceptional points" must all be points of discontinuity of $f$. Thus, for examples:

*

*the normal distribution with its usual density has no exceptional points

*the exponential distribution with its usual density has $0$ as an exceptional point and no others.

