Fly leaders are tapered monofilament strands that connect the fly fishing line to the fly. They're designed to cast (present) flies the way the fish expects to see food items.

Fly Leader Sections

Fly leaders have three sections: butt, midsection and tippet. Fly leaders are tapered from the thick, heavy butt to the narrow, thin fly tippet to help turn the fly over in casting. Fly fishing leaders are either hand-tied, with segments between knots; or they are knotless with a continuous taper from butt to fly tippet end. The packaging will give you specific information about the fly leader and how it will perform.
 
Tippet Tippet

What Fly Leader to Use

The right fly leader is the one that suits the fishing situation. For example, when you're fishing small flies on flat water for trout or panfish, the fly leader should be small in diameter. The fly leader must have a relatively soft fly tippet for soft presentation of the fly, and it must be long enough to assure that the fish is not spooked by the line hitting the water during the cast.
In general, the smaller the fly, the smaller the fly leader diameter at the tippet (end); the larger the fly, the larger the fly leader diameter. Also, the smaller the fly, the flatter the water and the more easily spooked the fish are, the longer and thinner the fly leader needed to fool them. Some spring-creek trout are so finicky that you may have to go lighter, while on salt water the fly tippet may have a breaking strength of 20 pounds with a shock tippet of 100 pounds tied in.
Fortunately, leader manufacturers have labeled their products to help you choose the right fly leader for your fishing. For instance, a bass fly leader should be relatively short and stiff to turn over the large flies you will use. Saltwater fly leaders will be relatively stiff, strong and very abrasion-resistant to withstand the large, sharp teeth of saltwater fish and the powerful runs made by those fish.
Fly Fishing Tippets

Connecting Fly Leader and Fly Line

The connection you make between your fly leader butt and the fly line is important. If you decide to use a knot, the easiest is the tube or nail knot. The needle nail knot is the most secure of all the leader/line connections, but it takes more time to tie. (The needle/nail knot connection is recommended for saltwater fishing.) A good fly shop will set up your line and leader for you.

The Right Leader Length

A standard fly leader for stream trout, bass and panfish is 9 feet. Some fly leaders may run as long as 22 feet, and spring-creek fly leaders are often 12 feet or longer. As the wind rises, casting these leaders becomes more and more difficult, so experienced fishermen shorten the fly leader to aid in turning the fly over in the wind.