What is a directed acyclic graph (DAG)? I am reading this link on Wikipedia; it states the following definition is given for a DAG.

Definition: A DAG is a finite, directed graph with no directed cycles.

Reading this definition believes me to think that the digraph below would be a DAG as there are no directed cycles here (there are cycles of the underlying graph but there are no directed cycles).

However, all the pictures on Wikipedia show examples of DAGs with arrows pointing the same way. So, I think I am interpreting this definition wrong. In particular, why does the definition mention later on an equivalent definition is that it must have topological ordering such that "every edge is directed from earlier to later in the sequence"? Reading the definition above would lead me to believe that the graph above is a DAG, but then the equivalent definition would make me think otherwise. 
 A: The graph you show is a DAG.
It is conventional to draw DAGs with all the arrows going in the roughly the same direction, because that usually gives a clearer intuition about what is going on in the graph.
But remember that locations and directions are not part of the formal definition of a graph -- they're just incidental features of the particular drawing at the graph you're looking at, and it would be the same graph if you drew the vertices in different locations on the paper.
(Even in your drawing, all the edges go in a broadly southeasterly direction -- or at least more southeast than northwest -- so you're actually following the convention).

In particular, why does the definition mention later on an equivalent definition is that it must have topological ordering such that "every edge is directed from earlier to later in the sequence"?

Because that is another way to define the same class of graphs, and sometimes (but not always) a more productive way to think about them. You should be able to prove that the finite directed graphs that have no directed cycles are exactly the same as the finite directed graphs that have a topological ordering.
A: the given graph is indeed a DAG, 
The equivalent definition says that a graph $(V, E)$ is a dag if and only if you can find a total order that extends the order given by $E$. 
In simpler terms, let $u_1, \ldots, u_n$ be the elements of $V$ (the vertices), then $(V, E)$ is a dag if and only if you can find an order $<$ such that if $(u_i, u_k)\in E$ then $u_i < u_k$.
A: Both answers so far state that what you drew is a DAG. However, it is not a DAG by one definition in common use, because it has multiple edges between the two leftmost vertices. It is common to define a directed graph to be a pair $(V,E)$ where $V$ is a set, called the vertices, and $E \subseteq V \times V$ is a set, called the edges (excluding $(v,v)$ for all $v \in V$). A DAG is then a particular kind of directed graph (having no directed cycles). In particular, since $E$ is a set, there is no way to express the fact that there are two edges with the same starting and ending vertices (that would require a multiset). Therefore I would call what you drew a "directed acyclic multigraph". However, the reasoning for why how you draw it does not affect whether it is a DAG, as explained in Henning Makholm's answer, seems to have answered the question that you actually wanted to ask.
A: You can define things many ways as the Directed acyclic graph at: https://en.m.wikipedia.org/wiki/Magma_(algebra) under types of magma shows. You can define a graph multiple ways, Therefore, any type of graph has multiple definitions. Your drawing is a DAG under a definition of: a graph, whose vertices lack a cycle graph, when following the directed path ordered by direction.
