Because wool and hides were dyed in preparation for use in the Tabernacle, colouring or dyeing is prohibited on Shabbat.
The definition of dyeing involves the improvement or beautification of an object. The Biblical prohibition only applies when the colour is intended to remain permanently, but the Rabbis extended it to include colouring a surface which will quickly fade.

Painting or colouring of any kind is prohibited (sorry, kids, no crayons!).

The use of makeup comes under this subject. No makeup may be applied on Shabbat; including powder, nail polish, eye makeup, lipstick and hair dyes. One may however remove makeup (with a disposable tissue or paper napkin, and without the aid of a cream).

Changing something’s colour by making it dirty does not fall under the prohibition, although some rule that this falls under the Rabbinic prohibition (especially if the object used is a cloth). The major practical application of this is wiping blood from a wound.

It is preferable to use a disposable tissue or napkin to clean up blood, although if those are unavailable, a cloth may be used.
Adding colour to food or a beverage solely to improve its flavor is allowed. This permission does not extend to food meant only for decorative purposes (centerpieces containing coloured water).

One may not shine one’s shoes on Shabbat. One may walk or sit in the sun (and thus become tan) if one does not intend to tan.
Some rabbis rule that adding colour to water is prohibited, since it looks like making paint. Luckily, if one adds instant coffee to hot water already drawn from a vessel, one does not violate the prohibition.

Adapted from the Shabbos Home by Rabbi Simcha Bunim Cohen