Waves are vibrations that propagate (travel) through a medium. Examples: sound, water waves, earthquakes, light.
In a longitudinal wave, vibrations are in the same direction as the wave propagation. This causes compressions and rarefactions of the medium.
In a transverse wave, vibrations are at right angles to the direction of wave propagation. The amplitude of a transverse wave is equal to the height of a peak.

Animations of transverse and longitudinal waves here.

The speed of a wave is constant for a given medium.
Frequency is the vibration rate of the wave. It is measured in s^-1 (cycles per second) or Hz (hertz).
Wavelength is the distance along a repeating unit of the wave, for example from peak to peak or compression to compression. It is measured in m (metres).
Period is the time one wavelength takes to travel past a fixed point. It is measured in s (seconds).

Web page about amplitude and wavelength of a wave here.

An electromagnetic wave (light) is vibrations in the electric field at right angles to vibrations in the magnetic field. The way the waves twists around (or doesn't) as it travels is called polarisation. If for example if the electric field is always vibrating up and down, the wave is vertically polarised.
Visible light is only a small section of the electromagnetic spectrum, from about 400 to 700 nm wavelength (violet to red).
From lowest to highest energy (lowest to highest frequency): radio, microwave, infrared, visible, ultraviolet, X-ray, gamma.

Diagram of the electromagnetic spectrum here.

In case you're interested, a video about light made in the 1940s can be found on YouTube here.