Linea Algebra: I do not understand this definition of a Vector Space I'm studying linear algebra, I'm reviewing Vector Spaces, and i came across the definition of this Real Vector Space $( \mathcal P_n (\mathbb R), +, \cdot\mathbb R)$:

where the 'addition' and 'multiplication' operations are defined as follows:

And I just don't know how to read these definitions, i.e. what do they mean
(of both, the real vector space and the operations)

The source of your confusion may be this: the symbols $+$ and $\cdot$ [although suppressed] are being used in two different ways. 
First of all we have them in the set of polynomials, just the usual multiplications $a_j x^j$ and so on, and usual additions $a_0+a_1 x +a_2 x^2$ and so on.
But then we want to make this into a Vector Space, so we need to define Vector Addition, and Scalar Multiplication by real numbers. Let's call these operations $\dotplus$ and $\cdot$ for a moment. Then the statement is that 
$$
(a_0+a_1 x+\dots+ a_n x^n)\dotplus(b_0+b_1 x+\dots+ b_n x^n)$$
is defined to be
$$((a_0+b_0)+(a_1+b_1) x+\dots+ (a_n +b_n) x^n)$$
and
$$c \cdot (a_0+a_1 x+\dots+ a_n x^n)$$
is defined to be
$$(c a_0+c a_1 x+\dots+c a_n x^n).$$
You now need to check that if you do this then the nine (?) axioms for a vector space are satisfied. 
Finally, once you are comfortable with what the notation means, you can be a bit more casual. No confusion is introduced since, for example we have that $a_0 \dotplus a_1 x$ (note the plus with a dot) is actually the same by definition  as $a_0+a_1 x$.
