Paint is really only two things, pigment and binder.
You could think of pigment as tiny rock particles. In fact, sometimes that’s exactly what they are. A well made paint will have each of those particles surrounded with a binder/vehicle. I group the terms binder and vehicle because they’re often the same material. The binder is what holds pigment particles together and to the surface of the painting. The vehicle is what makes the paint usable, brushable, able to get from point A to point B. Water could be an acceptable vehicle but once the water evaporates, there’s nothing to bind the pigment so it just falls off like dust.
For oil paint, the binder is typically linseed oil but could be another drying oil – walnut, safflower, poppy. Not all vegetable oils are the same though. Most don’t have the fatty acids that allow the oil to form chemical bonds with oxygen that make a hard dry film. Olive oil or corn oil, for example, will never dry so they aren’t good for making paint.
Watercolor and gouache paint have gum arabic as the binder; acrylics have an acrylic polymer emulsion, egg yolk for egg tempera, etc.