Fluorocarbon is the newest of the three lines types you'll see on the market today. It has a high abrasion resistance, but not as high as braid. It has less stretch than mono, but more than braid. Unlike the other two lines – mono and braid – fluorocarbon sinks, so it isn't your best choice when fishing topwater.
Perhaps the most unique thing about fluorocarbon compared to the other two lines is the fact that it is virtually invisible in the water column. Fishing with fluorocarbon on bodies of water that are extremely clear is a must. I've fished braid and fluorocarbon, one after the other on clear bodies of water to see if the fish could tell the difference. There is a significate difference in the number bites you will get when using fluoro.
Fluorocarbon is probably the least forgiving of the three line types. It is very easy to damage the line when tying your knot, which commonly results in knot failure. I tie a clinch knot with fluorocarbon. There are better options out there, however the clinch knot I can tie quickly. The most important thing when tying the knot with fluorocarbon is to check your line above the knot. If you "burn the line," it will have a wavy appearance and you should re-tie. Wetting the line during the tying process will help you avoid this.
Fluorocarbon can have lots of line memory. I like to treat my fluorocarbon with line conditioner which helps with this problem. Once you've learned how to deal with the weaknesses of fluoro, it will quickly become one of your favorite fishing tools.