Fishing Rope

There are three main types of fishing rope available on the market today: Monofilament, fluorocarbon and braid. While all three will get the job done, applying the appropriate line to the appropriate situation will lead to more fish hooked up and more fish landed.
All fishing reels and rods were designed to hold a particularly sized line. Reels will list a set number of line poundage per yards (such as 6lb/210yds) and rods will list a line class (such as 6-10lb). What this is really referring to is not so much the line's pound test, but rather the line's diameter. However, these classifications were created when the majority of anglers were spooling their entire reel with monofilament line, which has been the primary line used by anglers for the last half century.

Monofilament
Monofilament is a buoyant, water-absorbent line that has long been used by anglers on traditional spinning, spincasting and casting reels for its cast-ability and availability at mainstream sporting-goods stores. Unlike fluorocarbon, monofilament line floats and has the most stretch-ability of the three types of line.
Due to it being relatively inexpensive, monofilament is typically used as a main line, meaning the entire reel is spooled with monofilament and the lure is tied directly onto that line. Because of the line's buoyant properties, it is great for topwater lures and as a leader when fishing off the bottom. Also, due to its stretch-ability, it is a great option for crankbaits because the fish can turn and swim with the lure allowing you time to get a good hook set. A fish is also less likely to throw a hook tied on monofilament line when it goes airborne. The stretch-ability is also helpful when it's cold or the fish are finicky and you need them to hold the bait just a second longer so you can get a hook set.
A disadvantage of monofilament line is that it has a lot of line memory when it sits on a reel, meaning if you try to throw a lure that is too light for the line class of the rod the cast will more than likely result in line twists (the most common line problem anglers face). How do you get the memory out of a monofilament line on a spinning reel? Just add water. Unscrew the drag clip off the top of the reel and pop the spool off the reel. Hold the spool under warm to hot running water and rotate to make sure all of the line is getting some direct contact from the water. Heating the line like this will remove nearly all of your memory before hitting the water.

Fluorocarbon
Fluorocarbon line is all the rage in today's fishing community. Up until the last 5-10 years, fluorocarbon line has been used primarily as a leader and not a main line due to its lack of cast-ability. However, improvements in technology have fixed that problem and anglers have taken advantage. Unfortunately, it is still about twice as expensive as monofilament line, which is why most still use it as a leader instead of a main line.
Advantages of fluorocarbon line are that its nearly invisible under water because it has about the same refractive qualities as water and though it stretches some, it's more sensitive than monofilament. It is a great option for finesse fishing techniques in clear water. Three feet is typically enough leader when connected to braided line. Anything longer will require casting your connection knot through the guides of the rod which can lead to abrasion and knot failure, but may be necessary when using light spinning equipment in super clear water or if the fish are finicky. It's really just up to what you as an angler are comfortable with.
If fluorocarbon has any disadvantages, it's that it sinks and may not be as abrasion-resistant as monofilament. This can negatively affect fishing topwater lures. It can also lead to line breakage when casting or fighting a fish after retrieving your line through or along hard structure like submerged wood and rock.

Braid
Braided line has been around longer than most anglers realize, but advancements in cast-ability and break strength compared to line diameter have made braid a favorite line selection of anglers.
Braided line cuts through grass and wooden structure better than monofilament and fluorocarbon line, making it an optimal line for casting into heavy cover.
Braided line has no stretch giving it superb feel for the angler in regards to contact with their lure or bait. But because of that lack of stretch and superior break strength, tying braid directly to your hook is a risky endeavor, especially for bank anglers.