Difference between Fock space and Hilbert Space I am beginner in QFT. I would like to know why there is a need of constructing Fock space for a  $N$-particle system? Why can't we represent this many body system just as the tensor product of Hilbert space itself? In short what is the difference between a Fock space and a tensor product of Hilbert space $H_n$?
 A: A Fock space is just one special construction of a Hilbert space. The basic idea is that the Fock space allows you to superpose tensor products of distinct degree. In other words, it allows you to make sense of expressions of the form $$|a\rangle+|b\rangle\otimes |c\rangle.$$
where $|a\rangle,|b\rangle,|c\rangle$ are one-particle states. From the quantum mechanical point of view, if ${\cal H}_0$ is the "one-particle Hilbert space", representing the states of a single particle, then the states of a collection of $N$ particles of this kind, with this $N$ fixed, form a subspace of the tensor product of ${\cal H}_0$ with itself $N$ times:  ${\cal H}_0\otimes\cdots \otimes {\cal{H}}_0$.
The Fock space allows you to superpose such states and hence allows you to have a state on which for every $N\in \mathbb{N}$ you have probabilities for the number of particles being $N$: speak differently, you are allowed to describe states on which the very number of particles is uncertain and becomes an observable with probabilities and mean values as any other observable.
A very transparent example where this would be necessary is in relativity theory. The relation $E = mc^2$ implies that given enough energy particles can be created and that particles can be destroyed. This makes relativistic quantum mechanics with fixed number of particles problematic and the Fock space picture helps quite a lot.
So the construction is to simply form the direct sum of all symmetric or skew-symmetric tensor powers of ${\cal H}_0$. This yields either the bosonic or fermionic Fock space: $${\cal F}_\pm ({\cal H}_0)= \bigoplus_{n=0}^\infty (\cal H_0)^{\pm \otimes n}$$
where $({\cal H}_0)^{\pm \otimes n}$ means in my notation to take the tensor product of ${\cal H}_0$ with itself $n$ times and symmetrize for $+$ or anti-symmetrize for $-$.
To answer your question the distinction between the Fock space and a tensor product of Hilbert spaces is simply that the Fock space is a direct sum of infinitely many tensor products of one Hilbert space with itself.
