Nylon (pros - light, easy to buy,, inexpensive, many different kinds from military grade to cheap. Cons - frays, easy to chew through, carries scent, short lasting)
Biothane (pros - many colors, weatherproof, mold and mildew resistant, does not rot, easy to disinfect, long lasting. Cons - if it has rivets you need to check for tearing, can be sliced, slightly heavier than leather depending on type)
Leather (pros - common, long lasting, durable. Cons - requires maintenance, stretches, rots, 2 colors, not weatherproof, can stain, heavier than nylon)
Rope (pros - thicker, easier to handle than a flat lead, great for field dogs since it resists fraying and picking up brambles. Cons - can still fray after a lot of use, can be chewed, heavy)
Bungee (pros - great for building drive, versatile, can be used to stop pulling. Cons - expensive, specific uses)
Snap - your most common type of attachment point. Durable. Works fine. Spring can be broken, gate can stick and allow dog to get off leash.
Alligator - heavy duty. Great for agitation. Has a little lock you need to flip. Lock can break.
FROG clip - heavy. Fail proof. Strong. Can get stuck in brambles. Heavy. Easy to use. Good choice for escape artists.
MASH Clip - light weight. Strong. Fail proof. My choice for escape artists if you don't want a heavy attachment.
HK Snap - push gate. Easy to use. Not a great choice for escape artists. Great for rapid deployments.
Snap shackle - pull release. Strong but should be used with caution. Great for rapid deployments.
Traffic/pull tab 6 inches to 3 feet. Used to keep the dog close in tight situations, or as an attachment on a prong collar to correct a dog when the leash is attached to flat collar.
Obedience tab Usually 2-3 feet. Thin and lightweight. Used to help proof off leash obedience such as heeling. Lightweight so the dog can't feel the weight of the leash and thinks it's not tethered.
Drag line 4-10 feet. Thin line with no handle. Used for puppies around the house so you can keep an eye on them.
Lead 6 feet standard leash.
Waist Leash Depending on the company making it, usually a 3-6 foot leash with a hook on the end (handle optional) to be attached to your waist via belt.
Slip lead Instead of a clip, there's a ring at the end, and you slid the leash through that to make an adjustable loop to close around the head.
Tactical leash 7-8 foot leash with a ring on the handle to allow you to loop the leash and wear it around your waist or chest to keep the dog close and hands free.
Long line 10-50 foot line of varying widths used for recalls, distance work, or tracking.
Tracking line 8-50 foot line, usually on the thinner side, used for tracking. Length depends on the sport.
There are leashes with built in handles which are essentially a leash and traffic tab built in.
My personal choices in leads vary with what I want to do. I prefer biothane because I hate the weight of leather, the width of it, and I feel like I don't handle it well. I also prefer my biothane sewn and not riveted as I've had many riveted leads fail on me. That's just personal preference. The nylon leads I have are military grade mil-spec nylon since I like the feel and durability of it the best. I have one rope drag line I take to the ocean and the rest is biothane.
I have in my arsenal: a general 6 foot biothane leash with ring on the handle for keys, 3 foot lightweight obedience tab, 6 foot slip lead, 10 foot nylon drag line, 15 foot rope drag line, 30 foot thin biothane tracking line, and 7 foot nylon tactical lead. I probably have more lying around somewhere.
(Tyson's Soul Twin)