Does this kind of graph have a name other than it is an directed graph?
Does it have a property or characteristics? Visually I see $15$ edges and $16$ nodes. I want to learn more about graphs, but especially this one. I left out the numbers in the graph, but each node (vertex) has an unique number.
What can be said about this graph only to have the visuals in mathematics?
Your digraph has the property that, if you start at any node and follow the arrows, you always end up in the same place. (For a finite digraph, that's equivalent to saying that it's acyclic and has a unique sink.) Such a digraph is called an in-tree or an anti-arborescence among other things.. With all the arrows reversed, it's called an out-tree or an arborescence etc. Is that what you had in mind?