I am looking for an "easytounderstand" reference for Neron Models. Specifically if I have a semistable family of elliptic curves over $Spec {O}_K$ , with generic fibre $E_K$ and special fibre $E_k$ , then $E_k$ is an $N$gon of $\mathbb{P}^1$'s. In this context, what is the Neron model of $E_K$? I guess what I am asking is for a geometric description of the special fibre of the Neron model for $E_K$.
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The place to look for this is Chapter 4 ("The Neron Model") of Silverman's book Advanced Topics in the Arithmetic of Elliptic Curves, specifically Theorem 4.6.1: the Neron model of an elliptic curve is obtained by removing the singular points from the minimal regular proper model. Thus in your case the connected component is a rational curve with two points removed: as a group it is $\mathbb{G}_m$, the multiplicative group. The component group here is cyclic of order $N$. 

