Why do we see different colours?

We have special cells in our eyes called cones that are sensitive to different colors, effectively sending electrical impulses to our brain which processes them as colour. The only way we know we are seeing the same 'colour' as every one else is through words or language and colour vision testing!

There are two parts to this answer - we see colours as light reflects from objects. That light ray then eneters the eye focused by refraction in the cornea and the lens hitting the retina. The retina at the back of the eye has specialised cells called rods and cones that trigger a trichemical reaction in the eye which is then interpreted by the eye where we 'see' the colour.