Kelvin-Helmholtz (K-H) instability at a magnetohydrodynamic (MHD) tangential discontinuity (TD) is studied by means of two-dimensional MHD simulation. The TD is of finite thickness with both magnetic shear and velocity shear across the TD. Our simulation results indicate that the nonlinear evolution of MHD surface waves at the TD depends on the fast-mode Mach numbers of the plasma flows on two sides of the TD in the surface wave rest frame. When the fast-mode Mach numbers on both sides of the TD are less than 1, the K-H instability can grow into vortices or kink-type surface waves, depending on the orientation of the ambient magnetic field and the plasma beta. When the fast-mode Mach number on either side of the TD is greater than 1, nonlinear fast-mode plane waves are developed from the ridges of the surface waves and extended distance from the TD. A theoretical model based on the fast magnetosonic Mach cone formation is proposed to explain the formation of these nonlinear plane waves. The Mach angle of the fast magnetosonic Mach cone as a function of Mach number, orientation of ambient magnetic field, and plasma beta is derived. The flaring angles of these nonlinear plane waves measured from our simulation results are in good agreement with the Mach angles predicted by the theoretical model. Applications of our results to the Earth's magnetopause and to the solar wind are also discussed.