A connected graph is a graph for which every two nodes are connected by some path.
For example, a tree must be connected by definition.
There are a number of different definitions for connectivity of a directed graph. Often,
we say that a directed graph is connected if for every pair of nodes $u$ and $v$, there exists
a directed path connecting $v$ to $u$ as well as a directed path connecting $v$ to $u$. A graph
having this property may alternatively be called strongly connected.