TITLE: Definition/Clarification of Graph Embeddings
QUESTION [5 upvotes]: Recently I started reading about graph embeddings, but I am unable to grasp its definition from Wikipedia. Can anyone explain this term with an example.

REPLY [7 votes]: A graph is just an abstract idea, whereas a graph embedding is an actual physical instance of a graph that has to be drawn (or embedded) onto some surface.

In order to understand what a graph embedding is, it helps to first define what a graph is.

Definition: A graph is a set vertices and a set of edges connecting those vertices. In a way we can think of a graph as a set of pairs of values, where each pair is an edge and each value is a vertex.
To give an abstract example consider the graph $G = \{\{1,a\},\{\text{blue},a\},\{1,\text{blue}\}\}$.

Note that this definition doesn't at all mention a drawing or a picture of the graph; it's just a list of pairs. A graph embedding is where we have to take a graph and actually draw a picture of it on some surface. For example, consider these three drawings of $K_4$ in $\mathbb{R}^2$ (a flat surface):

Typically we require that no edges cross in a drawing of a graph to call it an embedding. This because we want to be consistent with the topological definition of an embedding. So really only the second and third drawings of $K_4$ above are embeddings into $\mathbb{R}^2$. Because we can embed $K_4$ in the plane, we say it is a planar graph. And of course you can start talking about embedding graphs on other surfaces. For example here is a graph embedded on a torus:

Image originally from www.ics.uci.edu/~eppstein/0xDE/F24/3dtorus.png 
And then an obvious question arises:

Question: Given a surface, what graphs can be embedded on that surface?

We've seen above that $K_4$ can be embedded on the plane, but it turns out that $K_5$ cannot! In fact we have a complete characterization of all graphs that can and cannot be embedded on the plane. The question of which graphs can and cannot be embedded on a torus, though, is still unanswered.