What is simplest polynomial algorithm for PLANARITY?
I am going to describe an algorithm. I am not sure it qualifies as "easy" and some of the proofs are not so easy. 

First we break the graph into 3-connected components, as mentioned by Chandra Chekuri.


Break the graph into connected components.
Break each connected component into 2-connected components. This can be done in polynomial time checking for each vertex $v$ of each 2-connected component $G_i$ whether $G_i-v$ is connected. 
Break each 2-connected component into 3-connected components. This can be done in polynomial time checking for two distinct vertices $v,u$ of each 2-connected component $G_i$ whether $G_i-\{ v,u \}$ is connected. 


We have reduced the problem to checking whether a 3-connected component of the graph is planar. Let $G$ denote a 3-connected component.


Take any cycle $C$ of $G$. 
Pin the vertices of $C$ as vertices of a convex polygon. Put each of the other vertices in the barycenter of its neighbors. This leads to a system of linear equations telling the coordinates of each vertex. Let $D$ be the resulting drawing; it may have crossings. 
If $D$ has no crossings, we are finished.
Take the vertices $U$ in any connected component of $G-V(C)$. The restriction of $D$ to the induced subgraph $G[U\cup V(C)]$ should be planar. Otherwise, $G$ is not planar. Take any face $F$ in the drawing $D$ restricted to the induced subgraph $G[U\cup V(C)]$, and let $C'$ be the cycle defining $F$. If $G$ is to be planar, then $C'$ must be a facial cycle. (When $C$ is a Hamiltonian cycle, then $C'$ should be constructed using an edge.)
Repeat step 2 with C' instead of C. If the resulting drawing is planar, then $G$ is planar. Else $G$ is not planar.


Remarks: 


Arguing that Tutte's spring embedding gives a planar embedding is not straightforward. I liked the presentation in the book of Edelsbrunner and Harer, Computational Topology, but it is only for triangulations. Colin de Verdiere discusses the spring embedding in http://www.di.ens.fr/~colin/cours/algo-graphs-surfaces.pdf , section 1.4. A general reference is Linial, Lovász, Wigderson: Rubber bands, convex embeddings and graph connectivity. Combinatorica 8(1): 91-102 (1988).
Solving a linear system of equations in a polyomial number of arithmetic operations is easy via Gaussian elimination. Solving it using a polyonomial number of bits is not that easy.