Dry flies look like umbrellas fully extended.
Wet flies look like umbrellas opened but not extended.
Nymphs look like umbrellas all folded up. :-D
Dry flies (at least the traditional catskills) have stiff hackles that hold the fly up on the surface of the water. So they look like an umbrella. Of course, flyfishers have been at this a long time, so things have gotten trickier than that. Some flies float on top without the hackling. Or the hackling is not in the traditional place. Still, if the materials are stiff and don't absorb water readily, it's probably a dry fly.
A wet fly can look a bit like a dry fly, but the materials are not stiff. In fact, they absorb water fairly readily. Once they do, the materials move under water and can look "alive". Wet flies can be very effective, and you can fish them a number of ways. Fish them like bait, and you'll have a good chance of success.
A nymph is tied to look like a well... nymph of an insect. So no fancy wings or antennae. They are a body and legs. So they are compact and have very light soft hackling or rubber legs sticking out. Again fish them like bait to start. You'll learn other techniques along the way too.
Here's an article you may want to read before catching your first trout:
http://www.flyfisherman.com/content/releasing-your-fish
And remember to check temperatures this time of year. Find a stream that is less than 70 in temperature at around 5pm (the time when the stream is warmest.) Fishing early in the morning may be best this time of year. That and an hour before and after dark.
