Information about colour is gathered by cells called cones in the eye, which communicate directly with neurons. Each cone responds to a different frequency range of light. In most cases, colourblindness happens when cones that are supposed to capture different ranges of colour have too much overlap. This does not mean that these people see in black and white! It just means that the brain has a hard time telling the difference between two shades, like red and green.
There are now special glasses that eliminate the areas of overlap and allow people with colourblindness to see more normally. The glasses modify the light spectrum in a way that the brain can interpret more easily. Some people have four cones rather than the usual three, which is called tetrachromacy (from “tetra”, meaning four, and “chroma”, meaning colour). They can differentiate colours that we cannot. Some animals naturally have 4 cones as well, and can see colours in the ultraviolet range!