What is the difference between a transverse wave and a longitudinal wave with examples?

The difference is in the direction the particles of the medium vibrate compared to the direction the wave travels. In a transverse wave, the particles vibrate at right angles (perpendicular) to the direction of wave travel, forming crests and troughs. Examples are waves on a string, ripples on water, and light and other electromagnetic waves. In a longitudinal wave, the particles vibrate back and forth along the same direction the wave travels, forming compressions (particles close together) and rarefactions (particles spread apart). The best example is sound travelling through air. A simple test: if you can picture crests and troughs, it is transverse; if you picture squeezing and stretching along the line of travel, it is longitudinal. Note that sound needs a medium, while light can travel through vacuum.