Edge of a graph
Elementary
Imagine a network drawn as dots connected by lines, like a map of cities and roads. Each line connecting two dots in the network is called an edge.
Undergraduate (graph theory)
Consider a graph, made of a set of vertices and a set of edges. An edge of the graph is an element of its edge set. In a simple undirected graph, an edge is an unordered pair $\lbrace u, v \rbrace$ of distinct vertices.
Notes
A graph edge has no position or length. Drawing it as a line segment is only a convenience.