Definitions:

A wave is energy conveyed, then transmitted by a medium from a point that has undergone a disturbance to another point.

Example:

When we talk, we disturb the air (medium) around us. This disturbance is conveyed and transmitted from us to another point. Such a disturbance is called the sound wave.This wave is propagated from us and travelled through the air which is the medium conveyor.

This example relates a wave that propagates through a material medium. It's then called a mechanical wave. As it travels, It's called a travelling wave; That's also the case of a water wave that propagates accross the surface of water or waves on ropes and strings. By contrast, a standing wave is confined in space by boundaries. We find standing waves in music instruments such as a guitar. Mechanical waves can be described with Newton's laws.
When a matter is disturbed, the resultant wave on it travels, but the particles of this medium do not move; they just oscillate. When the oscillation is "up-down", the resultant wave is transverse that shows crest and troughs. When the oscillation is "back-forth", the resultant wave is longitudinal that shows compression and expansion. For the transverse wave, the particles oscillate perpendicular to the propagation direction.For the longitudinal wave, the particles oscillate parallel to the propagation direction.
Some wave have both transverse and longitudinal components such as water waves. A water wave propagates radially outward from the point of the disturbance forming circles. Each of these expanding circles is alled a wavefront . The water wave propagates in two dimensions. It is an example of the plane waves .
Like the sound waves; the electromagnetic waves that are light, x-rays, radio waves and microwaves obtained by the oscillation of an electric field; propagate in three dimensions (all directions). In a uniform medium, their wavefronts form spherical surfaces centred at the the point source. They are called spherical waves.