I read through Wikipedia and tons of questions of MSE, but I still can't grasp the concept of a split of a short exact sequence.
Apparently by definition a sequence
$$0\rightarrow A\overset f \rightarrow B\overset g\rightarrow C \rightarrow 0$$
splits, if there exists a function $h:B\rightarrow A$ such that $f \circ h = 0$ or a function $t:C\rightarrow B$ such that $g \circ t = 0$.
Now in my book I have as a non-splitting example $A=2\mathbb Z/12\mathbb Z$, $B=\mathbb Z/12\mathbb Z$ and $C=\mathbb Z/2\mathbb Z$.
Furthermore $f(x+n\mathbb Z)=x+n'\mathbb Z$, where $x\in \mathbb Z$.
If I interpret the ideals or how they're called correctly, then for $h$ I would intuitively (and with simplified notation) say that it's
$$h:\{0,1,2,3,4,5,6,7,8,9,10,11\} \rightarrow \{0,2,4,6,8,10\} $$ and $$h(x+n'\mathbb Z)=x+n\mathbb Z,$$
so I'm "sending" the elements back to where they came from, but as I said, I still have no idea why this would be called a split (why that word, in particular), even if $h$ did exist.
Completely wrong understanding. You need $h$ so that $f \circ h$ is the identical map of $A$, and the same for the other one - it should be the identical map of $C$. And they must be homomorphisms, it is not enough for them to be "functions."