There are a lot of different fly lines out there made for the varying conditions, locations and fish you're chasing. You probably want to line your rod up with the proper line that'll make it the weapon of mass destruction you envisioned when you purchased it. There are two main styles of fly lines: weight forward and presentation (or double taper) lines. There's also full sink, sink tip, and lines for different species with attributes to target fish in the water columns they are likely to be in.
Weight Forward Lines
On the Madison we're typically fishing a weight forward (WF) or presentation line. A weight forward line is going to have a heavier tapered head on the front, allowing you to turn over big flies: streamers, nymphs, big dry flies like the salmonfly. One of our favorite WF lines is the SA Amplitude MPX, which is a half size heavier and gives you a bit more weight on the tip for turning over and casting into wind. The MPX is a textured line and great for fast action rods especially when fishing out of drift boats on large western rivers.
Textured lines cut through the air, lift off the water really well, and last longer. When fishing a textured line, you may notice that it can be a little more noisy than a smooth line when it's shooting through the guides.
Presentation or double taper lines are more traditional lines made for slower action presentation rods for spring creeks or targeting rising trout with smaller dries. A presentation line will often have a double taper (DT), but may be WF with a trout taper. Double taper lines will have a taper on both sides of the line, meaning you could flip the line and get double the life out of it.
Double taper lines have a lighter tip and allow you to present a dry fly really well. However a DT line will be more difficult to cast into wind or short distances.
There's all kinds of different lines out there; what I like to tell entry level anglers about to fall down the rabbit hole is that I'd rather see you spend more money on the line and less on the rest of your set up. Lines nowadays are so technologically advanced and have the most potential to affect your day on the river. Tight lines!