Is there a category-theoretic generalisation of function application? (I'll be using functional programming jargon, because my category theory is not too good).
Is the notion of function application, i.e. the application f a of a function f :: a -> b to a value a :: a to yield a value b :: b, peculiar to functions, or is there a more general category-theoretic concept to generalise it?
 A: (This should be rather a comment, but I will post it as an answer as it is too long)
Besides exponential objects as mention in the comments there is another way of 'evaluating' functions/morphisms:
In a general category we replace the elements (in the set-theoretic sense) with morphisms:
Let $\mathcal{C}$ be a category and $X$ an object in $\mathcal{C}$. We will call a morphism $T\to X$ a "generalized element of shape $T$" or (in algebraic geometry) "$T$-valued point of $X$". Now if you have a morphism $f:X\to Y$ and a generalized element $a$ of shape $T$ of $X$ we get an element $f(a)$ of the same shape of $Y$, i.e. $$f(a):=f\circ a$$
In the category of sets we know that two maps (with the same source and target) are equal iff they agree on elements (in the set-theoretic sense). Now this carries over to these generalized elements. Two morphisms $f,g:X\to Y$ are equal iff $f(a)=g(a)$ for all objects $T$ and generalized elements $a$ of shape $T$ of $X$.
As an example: In the category of sets we can identify the usual elements with these generalized elements of a certain shape $T$, in this case take $T$ to be your favorite set with one element. It is similar in the category of groups: here the usual elements of a group $G$ correspond exactly to the morphisms $\mathbb{Z}\to G$, that is to the generalized elements of shape $\mathbb{Z}$ of $G$.
A: Functions are morphisms in the category of sets. A morphism  $$e:S\rightarrow X$$ from any singleton S set to another set X (object in category Set), "picks out" an element of X. The composition of a morphism $$f:X\rightarrow Y$$ with the morphism e, $$f\circ e$$ "picks out" the value of f applied to the "value" of e.
