probability density function In a book the following sentence is told about probability density function at point $a$: "it is a measure of how likely it is that the random variable will be near $a$." What is the meaning of this ?

It would be illustrative to compare the case with a discrete R.V. With a discrete R.V, the pdf gives the distribution of the probabilities at specific points i.e, it gives you the probability that the R.V takes a particular value. With a continuous R.V, the pdf gives the distribution of probabilities over an interval i.e, it gives you the probability that the R.V takes values in an interval.
Mathematically, if $f(x)$ is the pdf of a discrete R.V - X, then $f(x) = P(X = x)$. If $f(x)$ is the pdf of a continuous R.V - X, then $f(x) = P(X \le x)$.
If $f(x)$ is the probability density function of a continuous R.V, then, $f(x)dx$ is the linear approximation to the cumulative probability of $X$ over the interval $[x - dx, x]$. (If you integrate the pdf, then as Colin Bowers said earlier, you get the cumulative distribution function, which gives the exact cumulative probability over any arbitrary interval $[x - dx, x]$.
It must also be noted that the 'differential probability' - $f(x)dx$ also depends on the interval $[x - dx, x]$ and not just $f(x)$. How small the interval should be, depends on how rapidly f(x) fluctuates.If $f(x)$ varies very slowly, you can take a larger neighbourhood, while if it varies rapidly, you would have to consider a smaller neighbourhood(for the linear approximation to be reasonably accurate).
