What Paint Is
Paint is composed of finely ground colored pigment, which is suspended in a medium, often water-based or a mix of water and oil. Paint is a suspension rather than a solution -- meaning that the pigment particles float in the medium, rather than dissolving in it. This is how paint gives something color: When the medium evaporates away, the suspended pigments are left behind on the painted surface.
Colloids and Emulsions
Most suspensions settle over time, but a subset of suspensions called colloids do not separate. Colloids can be made only with extremely fine particles in suspension. When the suspended particles are very small, as they are in paint pigments, the molecular forces acting on them prevent them from settling and give the suspension stability. Paints are a special kind of colloid called an emulsion. In an emulsion, tiny oily droplets are suspended in water. Water-based paints are made of oily pigments suspended in water. In oil-based paints, the pigments are suspended in tiny drops of linseed oil, which are then themselves suspended in water. When the paint is spread, both the water and the oil evaporate, leaving the pigment behind.
Molecular Interactions
The pigments in colloids and emulsions don't settle because of tiny interactions between the suspended particles. When a suspension settles, it's because the individual suspended particles bump into each other and stick together in a clump. As the clumps get bigger, gravity pulls them to the bottom and the suspension separates. In a colloid, like paint, the suspended particles repel each other. When two particles come in contact, they don't stick together then start to sink down, but instead repel each other and stay suspended. Because they are ground extremely fine, the pigments have an enormous total surface area, so the forces of repulsion are very strong. Instead of clumping together, the pigment particles are driven apart and disperse evenly throughout the medium.
Brownian Motion
The pigment molecules are in motion, bumping into the molecules of the suspending liquid constantly. They're so small that just the force of the molecular impacts is enough to keep the suspended particles in motion. The paint is always in motion at the molecular level -- in a sense, paint doesn't settle because it's stirring itself.