Dry fly
Designed to imitate aquatic or terrestrial insects floating on the surface of the river. Generally dry flies are cast upstream and allowed to drift back with the current.
The cool thing with dry fly fishing is that you usually see the trout eat your fly off the surface
Nymph
Generally designed to imitate the small insect larvae that live on the bottom of the river. Nymphs are generally cast upstream and allowed to drift back with the current.
Nymphs often have weight built into them with lead or beads to help them sink to the bottom of the river.
Streamer
Generally imitating baitfish.
Can be cast out and retrieved or cast across the river and allowed to swing downstream.
Streamers can be either weighted or unweighted. A floating or sinking line can be used.