I don't quite understand how I know if a polynomial is irreducible, and I would like to know if someone could help me understand that.
For example, if I have the following polynomial: $x^5+x^2+x^3+1$, is this polynomial irreducible in $\mathbb{Z}_{2}[x]$? I know that $\mathbb Z_{2}[x]=\{x^2, x^2+1, x^2+x, x^2+x+1\}$ but I will not continue from there to answer that question
In general, showing that a polynomial is irreducible is hard. There are a few irreducibility tests that can be used, but there are few general methods that don't require lots of computation or case-checking. One special case that you might have seen is that a cubic is irreducible over a field $k$ if and only if it has no roots in $k$; this is because any nontrivial factorization of a cubic must involve a term of degree at most $1$, and thus give a root. You can generalize this to higher degrees -- for example, for a degree $5$ polynomial, you only need to check all possible linear and quadratic factors -- but if your polynomial or field is very large this can get tricky.
However, showing that a polynomial is reducible is "easy" (easy in the sense that the data needed is short, not that it's necessarily easy to find). All you need to do is find a nontrivial factorization of your polynomial, and then you're done.
In particular, for the degree-$5$ polynomial you've written down, $x=1$ is a root in $\mathbb Z/2\mathbb Z$, so you can write it as $(x-1)$ times a degree-$4$ polynomial. This is a nontrivial factorization, and so the polynomial is reducible.