Paints are made by adding pigment (color) to something that will help it stick to the surface of a canvas (or anything else it can be painted on).
Color, also referred to as pigment, is a powder with a specific color. For artist grade and premium grade paints this is ground up metal in most cases. Cobalt blue is made with Cobalt powder mixed into a medium.
For Student grade paints, the pigment is usually obtained by using a colored die and mixing it with chalk (the one you write on a black board with) or marble dust. This helps manufacturers reduce the cost of the paint.
Oil paints are generally made by grinding the pigment powder with linseed oil. The grinding can be done by manually (with grinding stone, etc) or using a machine that can do it much faster.
Linseed oil has forms the most flexible film when it dries compared to other drying oils. The disadvantage of linseed oil is that it tends to yellow more than other oils over time. The other oil used for making oil paints is walnut oil, which doesn’t yellow as much but is forms a less flexible film than linseed oil does. It is, sometimes, mixed with linseed oil so the resultant paint will have a strong film and yellow less.
Cheaper oil paints tend to have less pigment and oil and more filler (marble dust, chalk or any other filler). Artists grade paints have less filler but they do contain some. Premium grade paints have no filler. They may, however, have stabilizers for certain pigments.
Acrylic paints are made by mixing pigment into acrylic emulsions (plastic). Like oil paints, artist grade acrylic paints have more pigment and than student grade paints.
The medium for watercolor paints is not water. It is gum arabic. Pigment added to gum arabic to make watercolors. Gum arabic is soluble in water which is why water can be used with watercolors as medium (to move the paint around, etc)
This is not as popular as the other 3 mediums but this has been around for a while. Egg tempera paints are made by mixing pigment into the egg yolk (not the egg white).
Encaustic paints are wax sticks that heated and the wax is used for painting while it is hot. When the wax dries it cannot be manipulated. Encausing paints are made by adding pigment to heated beeswax