Weight
Start by understanding that both fly rods and fly lines use a "weight" designation to indicate their size. In practical use, these designations don't have a lot to do with the actual weight of the line, but they are important in allowing you to make sure that you are matching your rod with a compatible line. Look at the printing on your fly rod right above the grip: this is where you'll find the weight designation There are very light-weight fly lines that are known as "zero weight," and lines available all the way up to a 15, but commonly used line weights lie between these two extremes. A fly line weight of 6 is often referred to as an "all-round" weight because it can be used for much of the trout and warm water fishing in the U.S. Salmon and steelhead fishermen generally use fly line weights between 8-10, and saltwater fishermen chasing bigger game use the 10-14.
Tapers
Fly line comes in a variety of tapers designed to perform in specific fishing situations. Double taper lines are tapered the same way at both ends and can be turned around if one end becomes worn. These lines are denoted by the letters "DT." Weight forward lines ("WF") are perhaps the most popular because the line carries a little more weight in the front section, in theory making them cast a little easier and farther. There is also a level line ("L") that is used for distance casting and specialty applications.
Floating and Sinking Lines
The fly line industry also uses letters to indicate whether a line is floating, sinking or some combination. Floating lines ("F") are used for dry fly fishing, and are still very useful for some wet fly and nymph fishing. Sinking lines ("S") are just that, sinking at a prescribed rate per second that is indicated on the fly line packaging. Full sinking lines can present some difficulties in picking them up and casting, so a hybrid line that has a floating line with a sink tip ("F/T") has been developed. This is great for fishing situations that don't require a full deep sink, because the butt of the fly line will float while the tip sinks down to the fish.