A complete graph of 4 vertices can be represented with a square and also with a triangle with a vertex in the middle. I'm confused if I should call the two graphs isomorphic or the same?
Also, can all planar graphs be represented with straight non-intersecting edges in 2D?
1) The definition of graph doesn't contain any indicators of shape. The drawing of a graph is only a graphical representation. These 2 graphs may be the same graph and be isomorphic (if they are the same set) or just be isomorphic. Isomorphic is a "safer" term, which should be used.
2) Yes, it is a theorem proven independently by K. Wagner (1936), I. Fáry (1948) and S. K. Stein (1951) and is known as Fáry theorem. Detailed proof may be found in G. Chartrand, L. Lesniak, Graphs & Digraphs, Wadsworth &Brooks/Cole, 1986 or newer edition (written with Ping Zhang).