From $p_{1},p_{2},p_{3}$ choose a basis $B$ for $W$ I'm very confused because we haven't got vectors or matrices here. So I really have no idea how to solve that task. I thought about converting these to one matrix somehow but it doesn't seem to work. This is no homework, it's a task from an old exam.

Let $W= \text{span}(p_{1},p_{2},p_{3}), W \subseteq R_{2}[x]$
$p_{1}(x)= 4x^2+3x^3$
$p_{2}(x)= 1+2x^2+3x^3$
$p_{3}(x) = 3-2x^2+3x^3$
From $p_{1},p_{2},p_{3}$ choose a basis $B$ for $W$ and state why $B$
  is a basis for $W$.

And what does this $R_{2}[x]$ mean?
I really hope you can give a detailled answer, I will also reward that answer with a bounty because I need to know how to solve tasks like that!
 A: The symbol $\mathbb{R}_2[x]$ usually denotes the space of polynomials (in the variable $x$ with coefficients in $\mathbb{R}$) of degree $\leq 2$. Since in your case, some of your polynomials are of degree three, I'll assume that this is a typo in the original question and that the intent was to write $\mathbb{R}_3[x]$.
Assuming that, you are given three vectors $p_1,p_2,p_3$ in some vector space $\mathbb{R}_3[x]$ and consider the vector space $W$ spanned by them. This is a vector space of dimension $3$ or less. To determine the precise dimension of $W$ and a basis, we need to understand whether the vectors $(p_1,p_2,p_3)$ are linearly independent and if not, choose a subset which is linearly independent.
In your case, $(p_1)$ alone is linearly independent because it is non-zero. Next, $(p_1,p_2)$ is a linearly independent list because $p_2$ is not a scalar multiple of $p_1$ ($p_1$ has no free coefficient while $p_2$ has a free coefficient). As for $(p_1,p_2,p_3)$, we can see that
$$ 3 - 2x^2 + 3x^3 = 3(1 + 2x^2 + 3x^3) - 2(4x^2 + 3x^3) $$
so $p_3 = 3p_1 - 2p_2$ is a linear relation showing that $p_3$ is dependent on $(p_1,p_2)$. Hence, $W$ is two dimensional with a basis given by $(p_1,p_2)$.
