By the definition from the book of Kenneth Rosen, we know that a graph consists of a nonempty set of vertices and a set E of edges. Each edges has either one or two vertices associated with it, called its endpoints. An edge is said to connect its endpoints.
From his definition, does it literally mean we actually can form a graph for which the edge has only one vertice on it?
Yes, it does. These so-called loops do indeed have some significant meaning. For example, given a planar graph, the dual graph can contain loops (even if the planar graph was without loops). It is also interesting for the direct product of graphs, see e.g. here.