The physical properties of light are intensity (or quantity), direction, wavelength (or colour), polarity and coherence.
Intensity is in simple terms the brightness of light, at least if you ignore that the human eye is more sensitive to some colours than others. The human eye is most sensitive to green light, then red and least sensitive to blue light. Green light with the same intensity as blue light is perceived brighter.
Light is also 'travelling' in a specific direction as a ray. It can be reflected when bouncing of surfaces and can be refracted at the boundary between two transparent materials (e.g. air and glass) and change direction. Most light sources emit light in all directions at the same time, e.g. the sun, fire or light bulbs (ignoring the shadow cast by the socket). Using reflectors and lenses, many artificial sources of light are built to emit light in a more or less specific direction as a beam of light, e.g. flash lights or car head lamps.
The wavelength of the light is what we perceive as colour. Longer wavelengths are perceived as colours on the red side of the rainbow and with shorter wavelengths, we go through the colours orange, yellow, green, and blue until we reach the shortest wavelengths we are able to see.
Polarity and coherence are properties of light, to which the human eye is not directly perceptible, but they play a role in many physics applications. Controlling the polarity of light is e.g. important in LCD displays and coherence is relevant for lasers and holograms.
I am not sure what 'quality' in your first list is supposed to mean. Contrast and hardness are not direct properties of light, but more related to how we perceive the interaction between different sources of light.
Contrast is seen if two adjacent objects are reflecting light with different intensities, as with one dark and one bright object. Some times the term contrast is also used for different wavelengths (colour contrast), e.g. the edge between a red and a green object may be perceived as 'contrasty' even if both objects have the same brightness.
Light seems hard and will throw distinct shadows if an object is only lit with light coming from one direction, e.g. outside on a bright sunny day, where the sun is the only significant source of light. Light coming from many directions is considered soft or diffuse, as opposite to hard. Outside on a cloudy day, the light from the sun is scattered by the clouds and will come from many directions, not only the single point of the sun, and you will see that objects are casting no, or only very weak and soft shadows.