Number of groups of order $p^n$, where $p$ is prime for $n=1$, it is cyclic. so, the number is $1$
for $n=2$, it is Abelian. so, the number is $2$
for $n\geq 3$, I don't know.
Can you recommend a book or link which can be helpful for understanding this?
Not just the result. I want to know the process of proving
 A: You should see "Enemuration of Finite Groups" by P. M. Neumann et al. 
The general problem does not seem to be approachable, and a tremendous effort is done mainly by M. F. Newman, E. obrien, B. Eick, M. R. Vaughan-Lee, Hans Ulrich Besche and many others, just for settling some special cases.
It is known that the number of groups of order $p^n$ (let us denote it by $f(p,n)$) depends also on $p$, not only on $n$.
The explicit formula of $f(p,n)$ is known just for very small values of $n$:
$f(p,3)=5$, $f(p,4)=15$ for $p$ odd, and $f(2,4)=14$.  
The situation become more complicated  for $n=5$, the exact formula is : $f(p,5)=2p+61+(4,p-1)+2(3,p-1)$ for $p>3$.
Still more and more complicated for $n=6$: $f(p,6)=3p^2+39p+344+24(3,p-1)+11(4,p-1)+2(5,p-1)$
for $p>3$.
A formula of $f(p,7)$, $p>5$ can be found as the main theorem in http://www.ukma.kiev.ua/~osp/GroupTheory/GroupsP%5E7/paper-p7.pdf 
You should see the small groups library for other particular values and more references (for instance http://www.icm.tu-bs.de/ag_algebra/software/small/).
A famous conjecture of G. Higman asserts that $f(p,n)$, when $n$ is fixed, is determined by a finite family of polynomials $(P_i)$, the choice of the polynomial $P_i$ (that is   $f(p,n)=P_i(p)$) depends on the residue of $p$  modulo some fixed integer $N$. Such a function is called PORC (polynomial on residue classes), so Higman's conjecture asserts that $f(p,n)$, as a function of $p$, is PORC. 
On the other hand, if we fix $p$, we have the following asymptotic formula  for $f(p,n)$ (due to Higman and Sims): 
$f(p,n)=p^{2/27n^3+O(n^2)}$.     
