The Errera graph is the 17-node planar graph illustrated above that tangles the Kempe chains in Kempe's algorithm and thus provides an example of how Kempe's supposed proof of the four-color theorem fails.
The Fritsch graph and Soifer graph provide smaller (and in fact the smallest possible) counterexamples.
A number of other embeddings (many of which are vertex-vertex and/or edge-vertex degenerate) are illustrated above.
The Errera graph has no planar unit-distance embedding (since it contains the 9-node triangular cupola unit-distance forbidden
graph), but a beautiful three-dimensional unit-distance
embedding can be obtained from two oppositely-oriented copies of a gyroelongated
pentagonal pyramid, i.e., a truncated regular
icosahedron with one vertex and adjoining faces removed, adjoined at their pentagonal
faces (E. Weisstein, Mar. 8, 2022). This is related to its being the dual graph of the 
-fullerene, which is one
of the three fullerenes on 30 vertices.
