Emulsifiers work by acting as a link between water molecules and nonpolar molecules, attracting the water with one end and the emulsified molecules with the other. They are made by bonding fatty acids, which attract hydrophobic substances, to a hydrophilic polar group, which attracts water molecules. Normally, substances like oil and water would separate out, even after thorough mixing, but emulsifiers keep the molecules evenly distributed.
Milk is an example of a naturally occurring emulsion, with milk fat kept evenly distributed, at least for a while, in water. It is also an example of a colloid, as are many emulsions. There is no essential difference between a colloid and other solutions, except that the dissolved particles in colloids are larger than in other solutions.
Emulsions are a form of solution, but the emulsifiers are the solute of the water, not the fats or oils they emulsify. In most emulsions, the separation of hydrophobic molecules from water is only delayed, not stopped completely. That is one reason why in full-fat milk, cream eventually rises to the top.
Another very common emulsifier is soap. Soap is a good cleaning agent because it can both dissolve in water and break down oils. Since skin has natural oils, soap ensures washing with water is effective.