The hybrid Cartesian/immersed boundary method is applied to fluid-structure interaction of a moving flexible foil. A new algorithm is suggested to classify immersed boundary nodes based on edges crossing a boundary. Velocity vectors are reconstructed at the immersed boundary nodes by using the interpolation along a local normal line to the boundary. For eliminating pressure reconstruction, the hybrid staggered/non-staggered grid method is adapted. The deformation of an elastic body is modeled based on dynamic thin-plate theory. To validate the developed code first, free rotation of a foil in a channel flow is simulated and the computed angular motion is compared with other computational results. The code is then applied to the fluid-structure interaction of a moving flexible foil which undergoes large deformation due to the fluid loading caused by horizontal sinusoidal motion. It has been shown that the moving flexible foil can generate much larger vertical force than the corresponding rigid foil and the vertical force can be attributed to the downward fluid jet due to the alternating tail deflection.