Longitudinal Wave Propagation
Below is an animation of a propagating longitudinal wave among a number of particles. One may observe that the particle motion is in the same direction as the wave is traveling (focus on the red colored particle). One may also visualize a longitudinal wavelength as the distance between consecutive condensations (local increases in density in the vertical direction), or as the distance between consecutive rarefactions (local decreases in density in the vertical direction). Also notice how the particles themselves do not travel from left to right with the wave, rather they oscillate sinusoidally back and forth about a static equilibrium position. Longitudinal waves propagate in gasses, liquids, and solids as particles collide into one another and cause those particles to collide into other particles and so on. Longitudinal waves are the so called "P waves" (P for primary because they arrive first) in a seismogram recording during an earthquake.
Shear Wave Propagation
Below is an animation of a propagating shear wave among a number of particles. One may observe that the particle motion is perpendicular to the direction of the wave travel (focus on the red colored particle). Notice that there is no visible condensations or rarefactions of the particles. One may more easily visualize a shear wavelength along the top edge of the animation as the distance between consecutive peaks (local upward shear motion), or as the distance between consecutive valleys (local downward shear motion). Again, the particles themselves do not travel from left to right with the wave, rather they oscillate sinusoidally up and down about a static equilibrium position. Shear waves mainly only propagate in solids and in some liquids; the particles in a solid can be thought of as being "attached" to one another by a series of springs (whereas this is not the case in gasses or most liquids). Shear waves travel at roughly half the speed of longitudinal waves in a bulk medium. Shear waves are the so called "S waves" in a seismogram recording during an earthquake.
Below is an animation of a rectangular solid bar being pulled and pushed up and down sinusoidally. As the bar is stretched vertically, the sides compress inward. As the bar is compressed, the sides expand outward. This effect is called the Poisson Effect. The ratio of the change in vertical length during an extension to the change in length horizontally during a corresponding compression is called Poisson's Ratio and is commonly 0.3 for many materials. Poisson's ratio is normally bounded between 0 and 0.5. For gasses and liquids, Poisson's ratio is effectively zero.