  In this paper, we contribute toward a classification of two-variable
polynomials by classifying (up to an automorphism of $C^2$) polynomials whose
Newton polygon is either a triangle or a line segment. Our classification has
several applications to the study of embeddings of algebraic curves in the
plane. In particular, we show that for any $k \ge 2$, there is an irreducible
curve with one place at infinity, which has at least $k$ inequivalent
embeddings in $C^2$. Also, upon combining our method with a well-known theorem
of Zaidenberg and Lin, we show that one can decide "almost" just by inspection
whether or not a polynomial fiber is an irreducible simply connected curve.
