The preferred terminology is vertex for a point and edge for a line. The lines need not be straight lines, and in fact the actual definition of a graph is not a geometric definition. The figure above is simply a visualization of a graph; the graph is a more abstract object, consisting of seven vertices, which we might name $\{v_1,\ldots,v_7\}$, and the collection of pairs of vertices that are connected; for a suitable assignment of names $v_i$ to the points in the diagram, the edges could be represented as $\{v_1,v_2\}$,$\{v_2,v_3\}$,$\{v_3,v_4\}$,$\{v_3,v_5\}$,$\{v_4,v_5\}$, $\{v_5,v_6\}$,$\{v_6,v_7\}$.