In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other.

A planar graph already drawn in the plane without edge intersections is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point in 2D space, and from every edge to a plane curve, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points. Plane graphs can be encoded by combinatorial maps.

Example of Planner graph

Butterfly Graph.

No, the complete graph of 5 vertices is non planar. because we cant make any such complete graph which draw without cross over the edges . if there exist any crossing with respect to edges then the graph is non planar.

Note:- a graph which contain minimum one edge from one vertex to another is called as complete graph...

Probably means 3 (as in 3dimensional).

(Bad terminology on maths' part it means a non-2d graph but 3d is still planar, just in 3 different dimensions)