What characteristics does P-waves and S- waves have?

P (Primary) and S (Secondary) Waves are Body Waves often associated with Seismic Waves as in Earthquakes.

Both Body Waves and Surfaces waves comprise an earthquake, but the body waves arrive first. Primary are first, And Secondary are, well, second of course.

P Waves are compressional which means they move through (compress) a solid or liquid by pushing or pulling similar to the way sound travels through the air. The particles of the material a P Wave pushes through move in the direction of the P wave's energy. This is called the direction of wave propagation.

S waves are slower than P waves and they can only travel through solid rock. S waves move the particles it pushes through up and down or side to side (perpendicular to the motion of the S waves energy).

The second link has very good animated illustrations of both movements.