David is correct. Different life stages in mayflies. They live most of their lives in the water.
Nymph - anytime. Represents the nymph, i.e. the aquatic bug before it begins transformation.
Emerger - just before or during hatches. Some mayflies molt from nymph to dun underwater, others as they're swimming to the surface, some at the surface, and even some swim to shore first and do it on the rocks! This imitates the nymph as its transforming, obviously its not very important for those that do it on the rocks.
Dun - During the hatch. After the duns get to the surface, they sit on the water and dry their wings before flying. As David said, they look like sailboats floating down, and if you see them in the air, they're usually going up. How long they stay on the surface often determines how much of a rise there is to them, and the emerger activity under water can be important too. It can be a function of species, or weather. Humid, or drizzly weather makes it tougher to dry their wings and they stay on the water longer, thus making more mayflies available to fish at any given time and usually making better fishing. Often its good to use the emerger as the hatch winds up and the dun as it winds down. Or you can do both, using a dun with an emerger dropper, which can be effective. BWO's are my favorite mayfly to fish the "dun" stage, the cool wet weather of March tends to mean they stay on the water a long time, and hatching activity is often quite concentrated right around the warmest parts of the day.
Spinner - After the duns fly off, they go to trees, molt into spinners (mating stage), this can take anywhere from minutes to hours. Then the mating flight begins, and usually en masse. All of the flies that hatched over the course of the day, or perhaps several days, do this at the same time. They dance up and down, starting high, and getting lower as they approach the water, and mate in midair to complete exhaustion. Then they fall on the water, dead or dying, with wings flat to the water (i.e. "spent"). You usually see the masses of mating mayfly's well before they hit the water in any sort of numbers, so you can often prepare some for these events. While the emergence shouldn't be overlooked, the spinner fall is usually the main event for species such as green drakes, sulfurs, slate drakes, and tricos, and its typically why us fly fishers want to be on the water at dusk and walk back to the cars with flashlights after dark.
Hatching activity usually lasts longer than a spinner fall. Spinner falls can be as short as a 20 minutes of fast action. Most spinner falls happen right at dusk.
Tricos are an exception. Emergence of the males happens at night, females right at dawn (just before and/or after sunrise). But the spinner fall is usually more important, it happens mid-morning. The exact timing depends on weather somewhat, but 8:30 or 9 is probably most common, right as the daytime sun starts to hit the water. I've seen it as early as 7 a.m. on hot sunny low humidity days and as late as early afternoon on rainy drizzly days.
