What is an emulsion and why do I care?

A photographic emulsion is the blend of chemicals that holds crystals of (typically) silver together in a coherent form so that when light is shined upon the surface of the emulsion, an image is formed instead of a slightly darker stew of chemicals. More simply, it holds all the photosensitive chemicals together in a convenient semi-gelatinous form.

The chemical definition of emulsion is a stable solution of one liquid in a second immiscible liquid--for example, whole milk, which has milkfats dispersed evenly in water. This is unusual because it is not typically energetically favorable to mix polar (water) and nonpolar liquids, for reasons that are beyond the scope of this writeup.

Although it's rather unlikely that you or anybody you know still uses the processes concerned, this word's relationship with photography began when the photosensitive matrix of chemicals that was exposed to light actually was an emulsion - the first process I'm aware of that used an emulsion in the chemical sense was the egg albumen print, which used (oddly enough) egg whites to suspend photographic chemicals on a glass plate.

Nowadays, talking about a photographic emulsion refers to whatever mix of gelatin and surfactants the nice people at the company have used to bind all the photosensitive chemicals together. This can mean a bottle of liquid emulsion, which is photosensitive and can be painted on any number of objects to photosensitize them. It can mean the layer of photosensitive chemicals and resin that coats RC printing paper, or it can (somewhat inaccurately) mean pretty much the entire film negative. The most inaccurate use is, appropriately, also the most common. It's not like you need to know much of this to start taking pictures, but without photographic emulsions, modern photography wouldn't exist.