A wave is a "propagating oscillation". A wave travels through a material when a force pushes on that material and the material resists being pushed. For example, when you speak, your voice compresses a volume of air. One of the properties of air (and just about any other material) is that it resists being compressed into a smaller volume. When your voice compresses this volume of air, its resistance pushes back against neighboring volumes of air. These volumes then resist compression, and they push back against their neighbors, generating a wave of compression that travels through all the volumes of air between the speaker and the hearer.
Earthquakes produce two types of body waves, called P- and S-waves (see P-wave). P-waves are sound waves. The particle motion in all sound waves is compressional which means that the particles move in the same direction as the wave as described in the example above. S waves are shear waves. The particle motion in shear waves is perpendicular to the direction of the wave.
Imagine a Slinky, held between two people. Pushing it sideways sends a wave down the Slinky*. As the wave passes through, the Slinky does not change its size, only its shape. If you let go of the top, it will return to its original, cylindrical shape, demonstrating the Slinky's resistance to shear. When one section of Slinky is sheared, its resistance transfers that deformation to the neighboring section, and a wave results.
Only a material that resists shear can transmit an S-wave. Thus shear waves are only seen in solids and cannot transmit through a liquid or a gas. The lack of S-waves through the outer core was one of the clues that showed Inge Lehmann that the outer core must be a liquid.
* Each piece of the Slinky moves back and forth sideways (this is the particle motion) while the wave travels the length of the Slinky (this is the wave direction.)