To go back to the very basics, a graph is made up of "nodes" and "edges".
If you want to draw a graph on a piece of paper, a
node would be a dot or a circle. An
edge would be a line connecting two nodes.
These pictures are all graphs. Notice, they're all just dots connected by lines.
Directed graphs are just graphs where instead of dots connected by lines, you have dots connected by arrows. You're correct in saying that a flow chart is like a graph. In fact, a flow chart is a graph, and a directed graph, to boot.
In a flow chart, actions and decisions are represented by the various shapes. These are the
nodes of the graph. You then have arrows that lead you from one action to the next, to the next, to the next. These arrows are the
edges.
Now, to explain directed acyclic graphs.
Acyclic just means there are no
cycles or loops in the graph. That means that no matter where you start on the directed graph, so long as you only follow the arrows, you will never pass over the same node twice.
Let me give you an example of an everyday process that is not acyclic
1. Wash
2. Rinse
3. Repeat
Notice the word, "repeat." That means "go back to step 1". This means do these steps in this order 1 -> 2 -> 3 -> 1 -> 2. That means there is a
cycle in this graph. There is a way to leave node 1 and eventually return by following the edges of the graph.
An example of a process that could be modeled using a DAG might be a tax form, like a 1040. These are formatted to follow in a very logical way. You simply do all the steps in order. Sometimes you will skip a step here or there, but you never go back and repeat a step.
