Difference between Probability and Probability Density
Simply put:
$\rho(x) \delta x$ is the probability of measuring $X$ in $[x,x+\delta x]$. With
$\rho(x):=$ probability density.
$\delta x:=$ interval length.
A probability will be obtained by computing the integral of $ \rho(x) $ over a given interval (i.e. the probability of getting $X\in [a,b] $ is $\int_a^b \rho(x) dx$. While $\rho(x)$ can diverge, the integral itself will not, and this is due to the fact that we ask that $\int_\mathbb{R}\rho(x) dx=1$, which means that the probability of measuring any outcome is 1 (we are sure that we will observe something). If the integral over the whole range gives 1, the integral over a smaller portion will give less than 1, because p.d.f. can't be negative (a negative probability is meaningless).
Probability density is a "density" FUNCTION f(X). While probability is a specific value realized over the range of [0, 1]. The density determines what the probabilities will be over a given range. What does it mean to have a probability density?
The probability density function for a given value of random variable X represents the density of probability (probability per unit random variable) within a particular range of that random variable X.
Now, I don't know how the p.d.f. can take value larger than 1
It is in this sense that probability density can take values larger than 1.
If you have a continuous random variable X with a value between 0 and 3 and the probability (is always between 0 and 1) that X will occur between 2 and 2.1 is say 0.2, the probability density (probability rate) will be 0.2/0.1 = 2. when you multiply the probability density by the interval of the event (2*0.1 = 0.2), you will get the probability.
The specific values $f(x)$ of the density function $f$ are the probability densities, and they express "relative probabilities", and the main point is that for a (measurable) subset $A$ of possible values (now $A\subseteq\Bbb R$), we have $$\int_Af\ =\ P(X\in A)$$ if the random variable $X$ has distribution described by $f$. In particular, $\int_{\Bbb R}f=1$, though its specific values, as shown by the given unlimited example, can be greater than $1$.