What does the value of a probability density function (PDF) at some x indicate? I understand that the probability mass function of a discrete random-variable X is $y=g(x)$. This means $P(X=x_0) = g(x_0)$.
Now, a probability density function of of a continuous random variable X is $y=f(x)$. Wikipedia defines this function $y$ to mean 

In probability theory, a probability density function (pdf), or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value.

I am confused about the meaning of 'relative likelihood' because it certainly does not mean probability! The probability $P(X<x_0)$ is given by some integral of the pdf.
So what does $f(x_0)$ indicate? It gives a real number, but isn't the relative likelihood of a specific value for a CRV always zero?

The ratio of the pdf $f(x)$ at two points, $r_x = f(x_0)/f(x_1)$, is not a measure of relative probability (or "relative likelihood") for the two outcomes for the random variable $X$. The ratio depends on the metric. That is, with a variable transformation, $z=z(x)$, with the pdf for $Z$ given by $h(z)$, the ratio $r_z=h(z_0)/h(z_1)\neq r_x$, in general, even though the two ratios refer to the same two outcomes. For monotonic transformations, $f(x)\,dx = h(z)\,dz$.
Numerical values of the pdf have no value on their own. The metric, $dx$, is required for probability interpretations (ie. $f(x)\,dx$). Wikipedia got this wrong, so I have corrected it.
