A functorial factorization is a structure on a category that factors any morphism into a composite of two morphisms, in a way that depends functorially on commutative squares.
Functorial factorizations play a prominent role in model category theory. On the one hand, many constructions there do rely on the factorizations into (acyclic) cofibrations and (acyclic) fibrations to be functorial, while, on the other hand, via the small object argument many examples of model categories do in fact carry a functorial factorization. (As a result, some authors include functorial factorization in the axioms of a model category right away.)
Functorial factorizations also play an important role as an ingredient in algebraic weak factorization systems.
A functorial factorization on a category $\mathcal{C}$ is a way of assigning to any arrow $f$ in $\mathcal{C}$ a pair of composable arrows $f_L, f_R$ such that $f = f_R \circ f_L$, together with for any commuting square
a morphism $E(h,k)$ completing their factorizations $f = f_R \circ f_L$ and $g = g_R \circ g_L$ to a further commuting diagram
in a way that depends functorially on the given commutative square, i.e. $E(h_1\circ h_2,k_1\circ k_2) = E(h_1,k_1) \circ E(h_2,k_2)$.
Write $\Delta[1] = \{0 \to 1\}$ and $\Delta[2] = \{0 \to 1 \to 2\}$ for the ordinal numbers, regarded as posets and hence as categories. The arrow category $Arr(\mathcal{C})$ is equivalently the functor category $\mathcal{C}^{\Delta[1]} \coloneqq Funct(\Delta[1], \mathcal{C})$, while $\mathcal{C}^{\Delta[2]}\coloneqq Funct(\Delta[2], \mathcal{C})$ has as objects pairs of composable morphisms in $\mathcal{C}$. There are three injective functors $\delta_i \colon [1] \rightarrow [2]$, where $\delta_i$ omits the index $i$ in its image. By precomposition, this induces functors $d_i \colon \mathcal{C}^{\Delta[2]} \longrightarrow \mathcal{C}^{\Delta[1]}$. Here
$d_1$ sends a pair of composable morphisms to their composition;
$d_2$ sends a pair of composable morphisms to the first morphism;
$d_0$ sends a pair of composable morphisms to the second morphism.
For $\mathcal{C}$ a category, a functorial factorization of the morphisms in $\mathcal{C}$ is a functor
which is a section of the composition functor $d_1 \;\colon\; \mathcal{C}^{\Delta[2]}\to \mathcal{C}^{\Delta[1]}$.
Not all weak factorization systems are functorial, although most are. This includes those produced by the small object argument, with due care, and also all algebraic weak factorization systems.
All orthogonal factorization systems are automatically functorial.
Sufficient conditions in enriched category theory (in particular enriched model category-theory) for functorial factorization to exist as an enriched functor is discussed in Hirschhorn 02, Theorem 4.3.8, Shulman 06, Prop. 24.2
The following are equivalent:
A endofunctor $R$ of $\mathcal{C}^{\Delta[1]}$ maps every morphism $f$ in $\mathcal{C}$ to a morphism $f_R$, in a functorial way. A point of such an endofunctor assigns to each $f$ a pair of morphisms $f_L, f_M$ such that $f_R \circ f_L = f_M\circ f$, naturally with respect to commutative squares. To say $cod \circ R = cod$ means that $f_R$ has the same codomain as $f$, and to say that $cod$ respects the points says that $f_M$ is an identity. Thus, $f = f_R \circ f_L$ is a factorization. For functoriality, the functoriality of $R$ gives the commutative square $k \circ f = f_R \circ E(h,k)$ and the functoriality of $E(-,-)$, while the naturality of $(-)_L$ gives the commutative square $E(h,k) \circ f = f_L \circ h$. The converse and dual are straightforward.
An algebraic weak factorization system is a functorial factorization together with compatible enhancements of these endofunctors to a monad and comonad. This can often be detected with the help of a composition law for factorizations.
Philip Hirschhorn, Model Categories and Their Localizations, AMS Math. Survey and Monographs Vol 99 (2002) (AMS, pdf toc, pdf)
Michael Shulman, Homotopy limits and colimits and enriched homotopy theory (arXiv:math/0610194)