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.