Four main different types of waves transport energy away from an earthquake.  Two of these move within the Earth and are called BODY WAVES  The other two types move along the Earth's surface and are called SURFACE WAVES.

To get an idea of these waves' motions, think about dropping a stone into a quiet pond.  The energy of the impact is transferred to the water (and air -- you hear it hit) and the energy is transported away from the impact as waves.  The water itself does not move away from the impact.  It simply rolls up and down as the waves pass -- only the energy is being transferred.  A quick glance at the waves suggests that individual wavefronts move in circular paths away from the impact, one followed by another.  However, this is not exactly accurate.  The initial wavefront, as it collapses, creates another wave immediately in front of itself, which creates another new wave in front of it, and so on.  the waves don't move, they are created one by one at the leading edge of the energy.  Thus, though we can talk about the wave movements it is really more accurate to say that the waves propagate away from the energy.  "To propagate" something means to produce new versions of it.  Rooting plant cuttings is also called "propagation" because you get new copies of the original plant.  The waves propogate in a circular fashion, allowing the energy to radiate away from the impact.

Though surface waves are obviously the ones that cause the damage and destruction associated with earthquakes, body waves are the more informative of the types.  The two types of body waves are shown in the diagram below.  In both cases the energy moves from the earthquake location off the left side of the picture toward the right, but the exact motions of the rocks the energy passes through are obviously different. 

In the upper wave the individual bits of rock move up and down (also in/out of he page, etc.) at right angles to the direction of wave propagation.  As each "bit" (each rock molecule or crystal) moves, it forces the one beside it to move as well, since they are attached.  These are called "s-waves", the "s" standing either for "shear" or "secondary" depending on who you ask.  (This type of particle movement, laterally beside each other, is called "shear movement".  Think of how the blades on a pair of "shears" move -- they go sideways past each other to propagate a cutting action at right angles to their motion.)  A rope tied to a wall and jiggled up and down is another good illustration of this type of motion.

In the lower part of diagram the particles can be seen to jiggle back and forth in the same direction the wave propagates, first bunching closer together (compression), then spreading farther apart (dilation).  These are called "p-waves", the "p" standing for either "pressure" or "primary" depending on who you ask.  A slinky or other spring is a good illustration of this type of motion.

Surface waves move in two ways that are similar to s-waves.  RAYLEIGH WAVES are essentially the same as deepwater waves in ocean water -- as the wave passes the surface makes a rolling vertical movement and ends just where it started -- rather like an s-wave going up and down, but with a little back-and-forth thrown in for good measure.  An object like a tall building will "whip" back and forth in the direction of wave motion each time a wave passes.  I had a Spanish professor once who was on the top floor of a tall hotel in Mexico City during an earthquake in the 1960's.  He said he knew what was happening when he awoke flying through the air toward the door and then suddenly found himself sliding back across the floor toward the bed.  A Rayleigh wave did this to him -- the building rocked beneath him.

LOVE WAVES are essentially s-waves that cause sideways motions of the ground as the wave passes.  These have a "twisting" effect on objects they pass under.