We consider the nonlinear graph p-Laplacian and the set of eigenvalues and associated eigenfunctions of this operator defined by a variational principle. We prove a nodal domain theorem for the graph p-Laplacian for any p >= 1. While for p > 1 the bounds on the number of weak and strong nodal domains are the same as for the linear graph Laplacian (p = 2), the behavior changes for p = 1. We show that the bounds are tight for p >= 1 as the bounds are attained by the eigenfunctions of the graph p-Laplacian of the path graph. Finally, using the properties of the nodal domains, we prove a higher-order Cheeger inequality for the graph p-Laplacian for p > 1. If the eigenfunction associated to the k-th variational eigenvalue of the graph p-Laplacian has exactly k strong nodal domains, then the higher order Cheeger inequality becomes tight as p -> 1.