Pulse:

Above is a demonstration of a pulse wave on a slinky, it is not a repeating wave and is mechanical.

Transverse Waves:

Above is a demonstration of  a transverse wave, the spring is moved to the side and let go and the wave goes across to the opposite side.

Periodic Waves:

Above is an example of a periodic wave, in which the wave is repeating at the same rate.

Longitudinal Waves:

Above is an example of a Longitudinal wave, the spring is pushed together at one end and the energy is released and a vibration goes parallel through the spring.