So now we know what an emulsion is: a dispersion of tiny drops of oil in water (it can also be the reverse, but why complicate things?). That dispersion won’t be stable unless we get a little chemical help to prevent all those little oil drops from coalescing together. That’s what emulsifiers are. They’re like bumpers, helping the individual drops to bounce off of one another instead of connecting and merging.
So how do they work? Emulsifiers are very interesting molecules in that they have both water-loving (hydrophilic) and fat loving (lipophilic) sections along their lengths. The technical term for a molecule of this type is an amphiphile. Introduce a whole bunch of amphiphiles to an emulsion and you can imagine what happens: the fat-loving sections of the molecules embed themselves in the oil drops with their water-loving bits sticking out. As the oil drops float around, they’re prevented from merging with one another by those protruding pieces, which hate fat and are literally repelled by it. The end result is that the oil drops remain suspended and the emulsion is stabilized.
So then are emulsifiers stabilizers? Of a sort, yes. However you can stabilize an emulsion to an even greater degree by adding actual stabilizers (starches, proteins or gums) to the emulsion. Why are they needed? Simply because, depending on how watery the emulsion is, fat drops can get forced together to the point that emulsifiers can’t keep them apart. Hence the need for some added interference.