A digraph is a graph where every edge is directed. How many digraphs on $n$ vertices are there? So far I have that between any two vertices (say $j$ and $k$) there are 3 options.


*

*there is no edge between $j$ and $k$

*there is an edge directed from $j$ to $k$

*there is an edge directed from $k$ to $j$


And so from this, we can see that there must be $3^{\binom{n}{2}}$ digraphs on $n$ vertices. 
I'm not sure if it should be $\binom{n}{2}$ or $\binom{n}{3}$? Can anybody clarify this for me?

As ever in combinatorics, you need to be crystal clear about what you're counting, and in particular about equivalence classes.
The OEIS sequence A000273: Number of directed graphs (or digraphs) with n nodes counts equivalence classes up to graph isomorphism and appears to exclude graphs with cycles of length 1 (as otherwise there would be two graphs on one vertex).
From your description you seem to be counting digraphs on $n$ labelled vertices without cycles of length 1 or 2.
So $3^\binom{n}{2}$ could be correct or not depending on what precisely the question is. $3^\binom{n}{3}$ is definitely not correct.
