Sorry for my sucky paint-drawing. The graph on the right have some extra properties than the one on the left. Doesn't it? It has a couple of two-way arrows and one vertex with a loop which the other one doesnt have. I know both are called Digraphs, but doesn't the one on the right have a more precise (better) name for it? to describe its properties? So I am wondering what it is called if I was to write about one or the other..
A digraph is called simple if it does not have any multiple edges (i.e., multiple directed edges with the same source and target vertices) or self-loops.
However, a simple digraph can still have "two-way edges", i.e., two edges of the form $(u,v)$ and $(v,u)$. Two such edges form a two-cycle.
So the graph on the left is a simple digraph with no two-cycles, while the graph on the right is a non-simple digraph with no multiple edges.