A Cartesian grid method is presented for solving elliptic equation on irregular domains with Robin boundary condition in this paper. The elliptic equation is reformulated into an elliptic interface problem on a larger regular domain, then solved by using the level-set immersed interface method (IIM) recently developed. In particular, the Robin boundary condition is discretized using one-sided cubic interpolation. The method is applied to solving the Navier-Stokes equations on irregular domains. The Navier-Stokes solver couples the ghost fluid method for the velocity equations and the IIM for the auxiliary variable equation. Numerical tests show that second-order accuracy is achieved in both solution and gradient for the elliptic solver, and with fast convergence. The Navier-Stokes solver produces second-order accurate velocity and one-order accurate pressure. The robustness of the Navier-Stokes solver is demonstrated through simulations of flow around a circular cylinder.