To be a Hilbert Space, your space $(V,+,\cdot)$ needs to:
Be a Vector Space:
$ \forall u,v,w\in V $ and $a,b\in\mathbb{F}$
1) Associativity of addition -- $u+(v+w)=(u+v)+w$
2) Commutativity of addition --- $u+v=v+w$
3) Identity Element of Addition -- $\exists 0\in V:\forall v\in V, v+0=0$
4) Inverse Element of Addition -- $\forall v\in V \exists -v\in V: v+(-v)=0$
5) Associativity with scalar multiplication -- $a(bv)=(ab)v$
6) Identity Element of Scalar Multiplication -- $1v=v$ for $1\in F$
7) Distributivity of scalar multiplication w.r.t vector addition -- $a(u+v)=au+av$
8) Distributivity of scalar multiplication w.r.t scalar addition -- $(a+b)v=av+bv$
Have an Inner Product
$\exists \langle\cdot,\cdot\rangle:V \times V \to \mathbb{F}$ with the properties:
1) Conjugate Symmetry -- $\langle v,w\rangle = \overline{\langle w, v \rangle}$
2) Linearity in the First Argument: $ \langle au, v\rangle=a\langle u,v\rangle$ and $\langle u+v, w\rangle = \langle u,w \rangle + \langle v, w \rangle$
3) Positive Definiteness -- $\langle v, v \rangle \geq 0$ and $\langle v,v\rangle = 0 \iff v=0$
Be Complete
If a series of vectors converges absolutely, i.e.:
$ \sum_{n=0}^{\infty} ||v_n||<\infty$
Then the series converges in the space V.
We can used spaces like this to model a vast array of physical systems. Sturm-Liousville Theory and Fourier Analysis rely on the properties of Hilbert Spaces of functions with an inner product defined using integration. These systems solve problems ranging from Classical to Statistical to Quantum Mechanics.
Now, by no means would I expect anyone to list why all of these properties are important, but what is it about some of the particular properties of Hilbert Spaces that makes them so useful?
For instance, why is Conjugate Symmetry of the inner product of two elements of the space necessary for describing so many systems? Or Linearity in the First Argument? For that matter, why should all Cauchy sequences of elements be required to converge in the space?
What is it about a complete, (usually) complex inner-product space that makes it so useful in so many ways?
P.S. (but not all that important): What exactly is meant when it says that a Hilbert Space generalizes the notion of Euclidean Space?