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A complete surface of Gaussian curvature zero in Euclidean three space is a cylinder (where a cylinder means the surface generated by the lines parallel to a given axis passing through a fixed curve in the subspace perpendicular to the axis; the plane is a cylinder in this sense when the curve is itself a line). This is due to Pogorelov, although it is usually attributed to Hartman-Nirenberg (they attribute it to Pogorelov). It can be found proved as Theorem 1 in W. Massey's Surfaces of Gaussian curvature zero in Euclidean $3$-space or in part II of Hartman and Nirenberg's On spherical image maps whose Jacobians do not change sign (Hartman and Nirenberg prove a more general result, for hypersurfaces in Euclidean space of arbitrary dimension). The papers of Massey and Hartman-Nirenberg contain more detailed results applicable to the incomplete case. Corollaries $3$ and $3^{\prime}$ of Hartman-Nirenberg show that every point of a $C^{2}$ surface of Gauss curvature zero has a neighborhood which admits a parameterization of the form $x = a(u)v + b(u)$ where $(u, v)$ varies over some simply-connected plane domain and $a$ and $b$ take values in $\mathbb{R}^{3}$. Probably by now more is known, particularly with respect to lower regularity. (See also chapter IX of the English translation Extrinsic Geometry of Convex Surfaces of Pogorelov's book.)