The kernel-cokernel exact sequence: in an abelian category, given $A \xrightarrow{f} B \xrightarrow{g} C$, the following sequence is exact

$$ 0 \to \ker f \to \ker gf \to \ker g \to \text{coker } f \to \text{coker }gf \to \text{coker} g \to 0$$

The maps are the obvious ones. The map $\ker g \to \text{coker } f$ is the one which factors through $B$.

I don't know if this fits, because it's not short and maybe it is too trivial, but I really think that every mathematician should know. For example, at a very low level, this tells the following basic facts

1. $gf$ is injective iff $f$ is injective and $\ker g \hookrightarrow{} \text{coker }f$
2. $gf$ is surjective iff $g$ is surjective and $\ker g \twoheadrightarrow \text{coker }f$
3. $gf$ is an isomorphism iff $f$ is injective, $g$ surjective and $\ker g \xrightarrow{\cong} \text{coker } f$
4. If $f$ and $g$ are injective/surjective, so is $gf$.

I think that more cools applications are covered in the following paper by Xiong, which I found just now.

A nice picture of this sequence from Nakaoka's website is the following