When you think of a wave, you may not necessarily think
of sound waves, or light waves, but instead ocean waves, or even a hand
waving. However, even these last two examples are useful in understanding
waves. A wave is a periodic disturbance in a medium (or in space). Both
the motion of the hand and of the ocean are periodic. The ocean wave is
a disturbance of the water, and the hand is a disturbance in space (you
could also say that it creates a disturbance in air!).
When considering wave propagation, there are two main kinds of waves, transverse waves, and longitudinal waves. Transverse waves are those in which the wave components (i.e. the individual parts of the medium that is transferring the wave) oscillate in a perpendicular direction to that of the wave motion. Consider a buoy sitting on the surface of the ocean, for example. As a wave goes by, the buoy rises with the crest of the wave, and falls with the trough. It bobs up and down regularly as the waves pass from one side of it to the other, but it doesn't get carried with the water. The motion of the buoy is in a vertical line, while the water moves horizontally. The crest of a wave is the highest point that it reaches, while the trough of the wave is the lowest point. These are respectively the maximum and minimum amplitudes, or displacement of the wave.
Next, consider a slinky on the floor, held by you and a friend. If you push your end of the slinky towards your friend, and pull it back towards you, a compressed section of slinky will effectively travel down to his end. If you had painted one loop in the slinky red, for example, what would happen to that loop as the compressed section traveled down the slinky? It would move towards your friend as the compression approached that part of the slinky, and away from him when the compression had passed. This kind of wave, where the components oscillate in a parallel direction to the wave motion is called a longitudinal wave. In this case, of course, the components of the medium that transmitted the wave were the loops in the slinky. Sometimes this kind of wave is also called a compressive wave, as it requires pressure on the medium in order to be propagated.
In a longitudinal wave, the crest and trough of a transverse wave correspond respectively to the compression, and the rarefaction. A compression is when the particles in the medium through which the wave is traveling are closer together than in its natural state, that is, when their density is greatest. A rarefaction is when these particles are further apart than is normal, or when their density is least.