"The purpose of drag mechanisms on fly reels is to provide braking pressure applied to the revolving reel spool by a mechanical device built in to the reel"
(Translated: a method of controlling the line so it doesn't leave the reel so fast.)

There are two basic drag systems used in fly reels:

Click & Pawl drag

Until the evolution of the now-ubiquitous disc drag, most fly reels used by the everyday fly fisher had the "traditional" click and pawl style drag mechanism.  (Also called "spring & pawl".)  They are characterized by having a spring-loaded "pawl" on the inside of the reel frame which engages a small "toothed" wheel on the center of the spool.  As the pawl on the frame ratchets against the teeth on the spool, the tension on the spring provides resistance to the spool as it turns to release line.

There are single pawl and dual/double pawl styles.  Some of both types are non-adjustable. . . you can't alter the pre-set drag.  Others can be adjusted, usually with a knob where you add tension on such reels to "tighten the drag."  Click & pawl reels are the ones that produce the "music" with which so many fly fishers become enamored.   It's the purring, clicking sound of the reel, as line is pulled off the reel or wound back on, that says "fly fishing!"

Disc drag

Reels with disc drags are currently the predominate style.  They are a "step up" in terms of their ability to provide resistance to a revolving reel spool which slows the speed of the fly line as it is taken off the reel by a running fish.  Reels using disc drags are heavier duty than click & pawl reels.  Most are tension adjustable via a knob on the frame. Generally speaking, a disc drag reel applies a more consistent level of restraint than a click & pawl system.  They therefore can handle a wider range of species of fish than can a click & pawl reel.

Disc drag systems use friction/compression between a metal disc and a piece of friction material.  They employ 1 1/2" - 2 1/2" diameter washers as the heart of the mechanism.  The most widely used material for such washers is cork, although many others are used: asbestos, Teflon, Delrin, rubber/cork composites, metal & leather.  Although arguments abound, experts seem to prefer the qualities that cork provides, probably because it has been around the longest.

For the most part, any production fly reel that you purchase on the market today, regardless of drag style, will be constructed of quality parts and materials, and the drag system is no different.

Let's evaluate the importance of the line and backing in fly reels: 

Drag to supplement the reel's built in mechanism can be also be applied by the fisherman in 2 ways:

1.  "drag" can be instantly increased or decreased by raising or lowering the rod tip . . . by raising it, we apply more restraint to the fly line;

2.  by "palming" a reel spool that has an "exposed rim", the fisherman can add more pressure to assist the reel's mechanical drag to slow down the turning spool and hence the line