An emulsion is the mixing of two different liquids that are immiscible, meaning they do not mix together to form a single phase. So the two liquids remain in separate phases as they are physically mixed together. One phase is often the dominant phase while the other is the dispersed phase (the phase that is broken up into droplets within the dominant phase). Oil and vinegar salad dressing is an excellent example. The two don't naturally mix, so shaking them together produced a temporary emulsified mixture.
"Emulsions should be examined after storage for droplet size of the dispersed phase." That is the phrase you asked about. This means that after forming an emulsion and letting it settle over time a bit, look at the dispersed phase (the phase that is broken into droplets) and measure the size of the droplets of the liquid that have formed. Different liquids will give different types of emulsions with different droplet sizes.