Given categories $C$ and $D$, the functor category – written $D^C$ or $[C,D]$ – is the category whose
morphisms are natural transformations between these functors.
Functor categories serve as the hom-categories in the strict 2-category Cat.
In the context of enriched category theory the functor category is generalized to the enriched functor category.
In the absence of the axiom of choice (including many internal situations), the appropriate notion to use is often instead the anafunctor category.
If $D$ has limits or colimits of a certain shape, then so does $[C,D]$ and they are computed pointwise. (However, if $D$ is not complete, then other limits in $[C,D]$ can exist “by accident” without being pointwise.)
If $C$ is small and $D$ is cartesian closed and complete, then $[C,D]$ is cartesian closed. See at cartesian closed category for a proof.
Functor categories enjoy the following accessibility and local presentability properties, as explained by Zhen Lin Low at nForum.
$\kappa$-accessible functors from a $\kappa$-accessible category to any accessible category form an accessible category. (It is not so easy to say what the accessibility rank is here.)
$\kappa$-accessible functors from a $\kappa$-accessible category to any locally $\lambda$-presentable category form a locally $\lambda$-presentable category.
Cocontinuous functors between locally presentable categories form a locally presentable category. More precisely, if $C$ and $D$ are locally $\kappa$-presentable, then so is $[C,D]$.
Continuous accessible functors between locally presentable categories form the opposite of a locally presentable category. More precisely, if $C$ and $D$ are locally $\kappa$-presentable, then so is $[C,D]^{\rm op}$.
Indeed, the point is this: given a $\kappa$-accessible category $\mathcal{C} \simeq Ind^\kappa (\mathcal{A})$ ($\mathcal{A}$ essentially small), the category of $\kappa$-accessible functors $\mathcal{C} \to \mathcal{D}$ (for arbitrary $\mathcal{D}$; here by “$\kappa$-accessible” we mean simply “preserves $\kappa$-filtered colimits”) is naturally equivalent to the category of all $\mathcal{A} \to \mathcal{D}$. It should be well known that:
If $\mathcal{D}$ is accessible, then so is $[\mathcal{A}, \mathcal{D}]$.
If $\mathcal{D}$ is locally $\lambda$-presentable, then so is $[\mathcal{A}, \mathcal{D}]$.
Colimit-preserving functors out of a locally $\kappa$-presentable category are $\kappa$-accessible.
A right adjoint between locally $\kappa$-presentable categories is $\kappa$-accessible if and only if its left adjoint is strongly $\kappa$-accessible (i.e. preserves $\kappa$-presentable objects as well as $\kappa$-filtered colimits); and every limit-preserving accessible functor between locally presentable categories is a right adjoint.
Statements 1 and 2 are proved in [Adamek and Rosický, Locally presentable and accessible categories], statement 3 is obvious, and statement 4 is a straightforward exercise. Thus the claims follow.
In general, accessible functors between accessible categories do not form an accessible category due to size issues. The best one can hope for is a class-accessible category. Let $\mathcal{C}$ be an accessible category that is not essentially small. Consider the category $\mathcal{A}$ of all accessible functors $\mathcal{C} \to \mathbf{Set}$. This is the same as the smallest full replete subcategory of $[\mathcal{C}, \mathbf{Set}]$ containing all representable functors and closed under small colimits. In particular, $\mathcal{A}$ is accessible if and only if $\mathcal{A}$ locally presentable. We claim $\mathcal{A}$ is not accessible.
Indeed, suppose $\mathcal{A}$ has a small generating family, say $\mathcal{G}$. Then for some regular cardinal $\kappa$, every member of $\mathcal{G}$ is $\kappa$-accessible. So consider $\mathcal{C} (X, -)$ for some object $X$ that is not $\kappa$-presentable. (Such an $X$ exists because $\mathcal{C}$ is not essentially small.) Since $\mathcal{G}$ generates, there is a small diagram of $\kappa$-accessible functors whose colimit is $\mathcal{C} (X, -)$. But then $\mathcal{C} (X, -)$ is a retract of a $\kappa$-accessible functor and hence $\kappa$-accessible: a contradiction. That said, $\mathcal{A}$ is a class-locally presentable category.
If $C$ and $D$ are small, then $[C,D]$ is also small.
If $C$ is small and $D$ is locally small, then $[C,D]$ is still locally small.
Even if $C$ and $D$ are locally small, if $C$ is not small, then $[C,D]$ will usually not be locally small.
As a partial converse to the above, if $C$ and $[C,Set]$ are locally small, then $C$ must be essentially small; see Freyd & Street (1995).