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?

'Relative likelihood' is indeed misleading. Look at it as a limit instead:
$$
f(x)=\lim_{h \to 0}\frac{F(x+h)-F(x)}{h}
$$
where $F(x)  = P(X \leq x)$
