Transverse Waves are waves with an outward motion of energy, and are transmitted through non-liquids. The reason for this occurs because in liquids, the molecules are moved in a non linear fashion, while Transverse waves are sent by the molecules it goes through moving along a single plane, but in the opposite direction of the wave.
Longitudinal Waves can be expressed in a spring. You pull one side and the potential energy being released into kinetic energy is done so in a wave, where the spring compresses as the wave passes through. The movement is again linear, but the molecules move in the same direction as the wave and move back to their original direction when the energy is gone.
Surface Waves, an example of which are tidal waves, are waves of energy that move the molecules in an interesting and less controlled manner. Instead of the particles within the media of the waves moving as they should, there are two medias in contest when the molecules are moving. This causes the molecules to swirl in a circular motion as energy passes through them.
Standing Waves are a rhythmic oscillation involving reflected pulses and superpositions. This basically means that the waves of energy that interact with one another on plane(s) create destructive and constructive paths as they make contact. The areas that are static are destructive, and the areas in motion are constructive (unless particles pass by the planar line made by the two wave starting points)