In physics, waves are disturbances or oscillations passing through a transmitting medium. Waves transfer energy, but they don't transfer matter.

A sine wave on a string. The blue particle moves up and down while  the wave travels to the right. The wave transmits energy but not matter. From wikipedia.

The above example is a one dimensional wave, but waves can be two dimensional (like ripples on the surface of water) or three dimensional (like sound waves in the air) as well.

Surface tension waves traveling through water. From wikipedia.

Waves have several properties that describe them, most notably: wavelength, frequency, amplitude and propagation velocity.  The wave's wavelength is  the length between two peaks (or troughs) on the wave. The larger the wavelength the more spread out the wave is.  The wave's amplitude is the distance of greatest displacement of the wave. The amplitude is associated with the intensity of the wave.

The wave's frequency is the number of times the wave oscillates in a given time.  The wave's propagation  velocity  is the velocity at which it propagates through the transmitting medium. (It is important to note that velocity is not the speed of the wave. Velocity is a vector. Its magnitude is the propagation speed, but it has a direction as well.) The wave's propagation velocity, frequency and wavelength can be related in the following formula:

v = λ*f        (velocity = wavelength*frequency)

The wave's propagation velocity is determined only by  the transmitting medium's properties.

One of the important concepts of waves is interference. Interference is simply the fact that when two waves meet, you can add the amplitudes of the two waves at the meeting point to find the real displacement. Constructive interference is when two peaks (or two troughs) meet, so the resulting wave is bigger than each wave; while destructive interference occurs when a peak and a trough meet, and the result is point of no displacement as the two cancel each other out:

Earlier we have met the theory that light is a wave. Thomas Young was able to show interference in light waves, as well as connected the lights wavelength to the color of light. Furthermore, he was able to explain the phenomenon of refraction through the change of propagation velocity when light passes into a different medium, as the same thing happens to other waves:

A plane wave enters a different medium where it travels at a slower velocity. As a result its angle changes - it is refracted. From wikipedia.

There was only one problem: if light is a wave, it must travel through a  transmitting medium. As light wasn't a regular displacement or pressure wave, and can travel through vacuum, there must be some  transmitting medium that fills all of space. This medium was called the luminiferous aether - a  light bearing medium. The properties of this medium need to allow the propagation of a wave of light.