Consider a pair of arrows $f: A \to B$, $g: A \to B$. An equalizer of $(f,g)$ is an arrow $e: E \to A$, such that $f \circ e = g \circ e$ and, for any arrow $h: D \to A$ such that $f \circ h = g \circ h$, there is a unique $\hat{h}: D \to E$ so that $h = e \circ \hat{h}$

So it is a morphism which makes $f$ and $g$ "equal", in a sense.

In Set, the equalizer for $f,g$ is given by the inclusion $\{x \in A | f(x) = g(x) \} \hookrightarrow A$.

Equalizers are Monic morphisms

They are unique up to unique isomorphism

Kernel of linear map f, is an equalizer of (f,0). We can define kernels in a similar way if we have the notion of a "0" map.

The Dual are Coequalizer, which generalize equivalence relations, and more general quotien structures..