There are many choices when it comes to the fly fishing line. Different types of tapering, floating/sinking, and many special lines. Let's go through the most common types.
A term that I use often is to "shoot". This means that every time your line goes forward or backward in the cast, you allow it to get a few more meters out. The more you can shoot, the faster you can get the line out and the further you can reach in the final cast where you shoot for the place you are aiming. Shooting is most important when you are casting long distances.
Double Taper, DT
A double tapered line has a thick body that evenly decreases in diameter towards both ends. A DT line is symmetric and can be reversed if necessary due to wear and tear, and that is an advantage. The disadvantage of the DT tapering is that it is difficult to cast really long with it since the thick middle will give more friction when you are trying to shoot with it to reach those last few meters.
However, it gives you good control and precision for short casting, so it is a good choice for short, precision casting e.g. in a small creek.
Weight Forward, WF
The WF line has most of the weight concentrated to the first 9 meters (30 feet) or so and after that, it quickly decreases in diameter. This means that once you have the entire belly of the line out, you have very little friction and can shoot for a long distance, much longer than DT.
The WF line is the most popular and the most versatile, and what I recommend for a beginner.
Shooting Head or Shooting Taper, ST
An ST is a special line that has a short heavy section in the beginning, normally around 6 meters (20 feet), and after that, it has a thin running line. Basically, this is just a more extreme version of the WF tapering.
An ST line is not for beginners, it is hard to use. The advantage of this line is that you can make really long casts with it, and therefore it is used in e.g. salmon fishing, saltwater fishing, etc. where this is necessary.
You can usually customize the exact length of the head to your own casting style, for better control.
Floating, sinking, or intermediate
Depending on the density of the line, it will either float, sink, or something in between.
The floating line is what you use for dry fly fishing, or for fishing near the surface with e.g. a nymph. It works for most types of fly fishing, and this is what I would recommend starting with.
To reach lower you need a sinking line. There are different speeds of sinking, from intermediate, that sinks very slowly, to lead core lines that sink fast.
The backing is what you have after the fly line to give you more length for when the fish rushes. It is usually made of dacron or sometimes nylon. Since the backing is all that protects your expensive fly line and your beautiful fish once you have the entire length of the fly line out, make sure you tie your knots properly and that the backing is not too old.