Propsition 1, part 1 (Maclane, CWM p.199)
Let $\mathsf A$ be an abelian category. Then every arrow has a factorization $f=me$, with $m$ monic and $e$ epic; moreover, $$m=\ker (\text{coker} f),\;\;\;\;e=\text{coker} (\ker f)$$
What is the intuitive meaning of this decomposition, in particular of the one consisting of the above $m,e$, and why is it significant?
It's easiest to understand with an example, so let $R$ be a ring and consider the category of left $R$-modules. If $f\colon M \to N$ is a homomorphism then categorically it's kernel and cokernel are the homomorphisms $$\mathrm{ker} \ f \to M \qquad \text{and} \qquad N \to N/\mathrm{im} \ f$$ where here $\mathrm{ker} \ f \subseteq M$ and $\mathrm{im} \ f \subseteq N$ take their non-categorical meanings as submodules (instead of their categorical meanings as maps).
Now the maps $m$ and $e$ are $$m\colon\mathrm{im} \ f \to N \qquad \text{and} \qquad e\colon M \to M/\mathrm{ker} \ f.$$ The first isomorphism theorem gives $M/\mathrm{ker} \ f \simeq \mathrm{im} \ f$ and the composition $$M \overset{e}{\rightarrow} M/\mathrm{ker} \ f \simeq \mathrm{im} \ f \overset{m}{\rightarrow} N$$ is in fact the map $f$.
As you see from the example, this is just factoring a map through it's image.