Let $M$ be a (fine) moduli space which parametrizes certain varieties (The moduli space in my mind is the moduli space of abelian surfaces with certain polarization -- but I don't know what is the moduli problem of this moduli space and whether it is fine or not). Then is there a variety $V$ over $M$ such that the fibre $V_s$ of closed point $s \in M$ is exactly the variety corresponding to the variety $s$ parametrizing?
I feel that universal bundle seems having something to do with $V$.
You should have a look at Chapter 8 of Lange and Birkenake Complex Abelian Varieties. I will refer to this book in what follows.
Fix a type $D= \textrm{diag}(d_1, \ldots, d_g)$. Then the moduli space of polarized complex abelian varieties of dimension $g$ and type $D$ with symplectic basis is precisely the Siegel upper half-plane $\mathfrak{H}_g$. So this moduli space admits a universal family, that is constructed in Section 8.7.
If you want to forget the symplectic basis, then you need to consider the moduli space $\mathcal{A}_D=\mathfrak{H}_g/G_D$, where $G_D$ is a suitable discrete subgroup of $\textrm{Sp}_{2g}(\mathbb{R})$. The space $\mathcal{A}_{D}$ is a normal, complex analytic space of dimension $\frac{g(g+1)}{2}$, which is called the moduli space of polarized abelian varieties of type $D$. Since the quotient of an algebraic variety under a good action is also an algebraic variety, it follows that $\mathcal{A}_{D}$ is a complex algebraic variety. However, it is not a fine moduli space, because of the existence of polarized abelian varieties with extra automorphisms, so one in general does not expect a universal family over $\mathcal{A}_g$.
In order to obtain a fine moduli space it is necessary to rigidify our objects. In the analytic category one can choose to consider symplectic bases, in the algebraic category the rigidification can be made by using the so-called level structures.
You can find (much) more details in the book by Birkenhake-Lange.
In general this is false. You are asking for a universal family. This is equivalent to say that the moduli problem is representable by a scheme.
For instace take $\overline{M}_{0,n}$ for $n\geq 5$ parametrizing $n$-pointed stable curves of arithmetic genus zero. Then $\pi:\overline{M}_{0,n+1}\rightarrow\overline{M}_{0,n}$ (the morphism which forget one marked point). Then the fiber of $\pi$ over a point $[C,x_1,...,x_n]\in \overline{M}_{0,n}$ is isomorphic to $C$. The morphism $\pi$ is the universal curve.
On the other hand if you consider $\pi:\overline{M}_{1,2}\rightarrow\overline{M}_{1,1}$ the fiber over $[E,p]\in\overline{M}_{1,1}$ is isomorphic to $E/Aut(E)\cong\mathbb{P}^1$ because any pointed elliptic curve has a non-trivial automorphism, namely the elliptic involution. In general there is no universal curve over $\overline{M}_{g,n}$. To represent this moduli functor and get a universal curve one needs a stack.
